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IIT – ian’s

P A C E

216 - 217 ,2nd floor , Shopper’s point , S. V. Road. Andheri (West) Mumbai – 400058 . Tel: 26245223 / 09

Practice Question Question based on

Q.1

LEVEL –1

Order and Degree of Differential Equation

(C) dy + dx = 0

(A) 1 Q.8

(B) 2

differential

(C) 3

equation

(D) 6

The order of the differential equation whose

(C) 1

(D) None of these

Q.9

The differential equation of all circles of radius a is of order(A) 2 (B) 3 (C) 4 (D) None of these

Q.10

The order of the differential equation of all circles

(B) 1, 1 (D) 2, 2

The order and degree of the differential equation 2

dy dy y=x + a 2 b 2 isdx dx

(A) 1, 2 (C) 1, 1

of radius r, having centre on y-axis and passing through the origin is(A) 1 (B) 2 (C) 3 (D) 4

(B) 2, 1 (D) 2, 2 Q.11

The order and degree of the differential equation dy 2 4 dx

2/3

=

4

– 4

(B) 3, 3 (D) 3, 2

d 3y

dx dx respectively(A) 4, 1 (C) 1, 1

3

+ 8

d2y dx

degree

d2y

dy +3 dx dx

are-

dx 2

The order and the degree of differential equation d4y

The 2

d2y

(A) 2, 2 (C) 2, 3

Q.6

the

solution is y = a cos x + b sin x + ce–x is(A) 3 (B) 2

The order and degree of differential equation

(A) 1, 2 (C) 2, 1

Q.5

of 3

(D) None of these

1 y 2 dx + y 1 x 2 dy = 0 are respectively-

Q.4

degree

2

d y dy (A) x – x + a = 0 (B) + xy = 0 dx 2 dx

Q.3

The

d2y dy + 1 = 0 is 2 dx dx

A differential equation of first order and first degree is2

Q.2

Q.7

Differential Equation

2

– 8

dy + 4y = 0 are dx

(B) 1, 4 (D) None of these

(A) 1 (C) 3 Q.12

of 2

the

differential

equation

d2y = x2 log 2 is dx

(B) 2 (D) None of these

The differential equation 2

4 d2y dy x 2 + + y = x2 is of dx dx

(A) degree 2 and order 2 (B) degree 1 and order 1 (C) degree 4 and order 3 (D) degree 4 and order 4

The order and degree of differential equation (xy2 + x) dx + (y – x2 y) dy = 0 are(A) 1, 2 (B) 2, 1 (C) 1, 1 (D) 2, 2

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1

Question based on

Q.13

Linear and Non-linear Differential Equation

Q.18

Which of the following equation is linear? (A)

dy 2 dy (A) y 1 = 2x dx dx

dy + xy2 = 1 dx

dy 2 dy (B) y 1 = 2x dx dx

dy (B) x2 + y = ex dx

(C)

dy + 3y = xy2 dx

(D) x

(C)

Which of the following equation is non- linear(A)

dy = cos x dx

(B)

Q.15

d2y 2

+y=0

Which of the following equation is linear ? 2

d2y (A) 2 + x2 dx

(B) y =

dx

2

(C) Q.20

dy =0 dx

dy dy + 1 dx dx

d y

2

dx 2

Q.17

dx 2

+ 9y = 0

d2y dx

2

d2y dx 2

–x

dy =0 dx

+x=0

(A) 2xy

dy + x2 = y2 dx

(B) 2xy

dy + x2 + y2 = 0 dx

dy + x2 = y2 dx

(D) None of these

(D)

d2y

Q.21

dx 2

The differential equation of all parabolas whose axes are parallel to y- axis is(A)

(C)

d 3y dx

3

d 3y dx

3

=0

+

Q.22

d2x dy

2

=0

(D)

d2x dy 2 d2y dx

2

=c

+2

dy =c dx

The differential equation of family of curve y = Aex + Be–x , where A and B are arbitrarily constants, is (A)

d2y dx

2

+y=0

2

dy + =0 dx

(B)

– 9y = 0

The differential equation of the family of curves represented by the equation x2 + y2 = a2 isdy dy (A) x + y =0 (B) y =x dx dx (C) y

(D)

dx

2

The differential equation of the family of curves

(C) xy

y = 4 sin 3x is a solution of the differential equationdy dy (A) + 8y = 0 (B) – 8y = 0 dx dx d2y

=0

d2y

represented by the equation (x – a)2 + y2 = a2 is-

Formation of Differential Equation

(C)

(B)

2

dy y + = log x dx x dy (D) y –4=x dx

Q.16

dy =0 dx

dy –x=0 dx

(C)

Question based on

+2

2

dy dy (D) + 3 +y=0 dx dx The differential equation of all the non-vertical lines in the xy- plane is-

(A)

dx dy 3 (D) x + = y2 dx dy / dx

(C) dx + dy = 0

d2y

3

dy + y2 = sin x dx

Q.19 Q.14

The differential equation of the family of curves y2 = 4a (x + a), where a is an arbitrary constant, is-

(D) None of these

(C) y

d2y dx 2

(B)

d2y dx 2

=y

2

dy – =0 dx

(D) None of these

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2

Q.23

Q.24

The differential equation for the line y = mx + c is (where c is arbitrary constant)(A)

dy =m dx

(B)

(C)

dy =0 dx

(D) None of these

Q.29

(D) y = ex cos x + c

Q.30

d2v

(B) y =

d2v

(C) y =

Q.31 Variable separable method

dy x = isdx y 2

(A) – = c (C) x2 + y2 = c

Q.27

Q.28

The solution of the differential equation

(A) y = tan (x2 + x + c)

2

y3

x2 log x – x2 + c 2

dy = (1 + x) (1 + y2) isdx

The general solution of the differential equation

x3

x2 +c 2

1 2 1 2 x + x log x + c 2 2 (D) None of these

2 dv (C) + =0 2 r dr dr (D) None of these

Q.26

dy = x log x isdx

(A) y = x2 log x –

2 dv (B) – =0 2 r dr dr

Q.25

The solution of

d2v

1 dv (A) + =0 2 r dr dr

Question based on

dy = ex (sin x + cos x) isdx

(A) y = ex (sin x – cos x) + c (B) y = ex (cos x – sin x) + c (C) y = ex sin x + c

dy +m=0 dx

The differential equation of the family of curves v A = + B, where A & B are arbitrary constants, isr

The solution of

(B) y = tan (2x2 + x + c) (C) y = tan (x2 – x + c) x3

y3

(B) + = c (D) x2 – y2 = c

The general solution of the equation (ey + 1) cos x dx + ey sin x dy = 0 is(A) (ey + 1) cos x = c (B) (ey – 1) sin x = c (C) (ey + 1) sin x = c (D) None of these The solution of the differential equation dy = sec2 x dx is(A) y = sec x tan x + c (B) y = 2 sec x + c 1 (C) y = tan x + c (D) None of these 2 The solution of the differential equation dy (1 + x2) = x isdx (A) y = tan–1 x + c (B) y = – tan–1 x + c 1 (C) y = loge (1 + x2) + c 2 1 (D) y = – log e(1 + x2) + c 2

x2 (D) y = tan x c 2

Q.32

The

solution

x sec y

of

the

differential

equation

dy = 1 isdx

(A) x sec y tan y = c (B) cx = sec y + tan y (C) cy = sec x tan x (D) cy = sec x + tan x Q.33

The solution of the equation dy 1 y2 + = 0 isdx 1 x 2

(A) x 1 y 2 – y 1 x 2 = c (B) x 1 y 2 + y 1 x 2 = c (C) x 1 y 2 + y 1 x 2 = c (D) None of these

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3

Q.34

Solution of the equation cos x cos y

Q.39

The solution of the differential (1 + cos x) dy = (1 – cos x) dx isx (A) y = 2 tan – x + c 2 (B) y = 2 tan x + x + c x (C) y = 2 tan + x + c 2 x (D) y = x – 2 tan + c 2

Q.40

If

Q.41

The solution of the differential equation sec2 x tan y dx + sec2 y tan x dy = 0 is(A) tan x = c tan y (B) tan x = c tan (x + y) (C) tan x = c cot y (D) tan x sec y = c

Q.42

The solution of

dy = – sin x sin y isdx

(A) sin y + cos x = c (B) sin y – cos x = c (C) sin y. cos x = c (D) sin y = c cos x Q.35

The

general

2

d y 2

=–

1 2

solution

of

the

equation

is-

dx x (A) y = log x + c1x + c2

(B) y = – log x + c1x + c2 1 + c1x + c2 x (D) None of these

(C) y = –

Q.36

Q.37

The general solution of the differential equation dy ey + (ey + 1) cot x = 0 isdx (A) (ey + 1) cos x = K (B) (ey + 1) cosec x = K (C) (ey + 1) sin x = K (D) None of these

x3 +c 3 (C) ey = ex + x3 + c

Q.38

x2 + log cos x + c 2

(B) y = ex + sin x +

x2 + log sec x + c 2

(C) y =

ex

x2 – sin x + + log cos x + c 2

(D) y = ex – sin x +

x2 + log sec x + c 2

= 0, then1 x 2 (A) y + sin–1 x = c (B) y2 + 2sin–1 x + c = 0 (C) x + sin–1 y = 0 (D) x2 + 2 sin–1 y = 1

dy 3e 2 x 3e 4 x = isdx e x e x

The general solution of differential equation dy (4 + 5 sin x) = cos x isdx 1 (A) y = log |4 + 5 sin x| + c 5 1 (B) y = log |4 + 5 cos x| + c 5 1 (C) y = – log |4 – 5 sec x| + c 5 (D) None of these

Q.44

The general solution of differential equation dy = log x is dx (A) y = x (log x + 1) + c (B) y + x (log x + 1) = c (C) y = x (log x – 1) + c (D) None of these

(D) None of these

(A) y = ex + sin x +

1

Q.43 (B) ey = ex + 2x + c

The solution of the differential dy = ex + cos x + x + tan x isdx

dy + dx

(A) y = e3x + c (B) –y = e3x – c (C) y = – e3x + c (D) None of these

The solution of the equation dy = e x–y + x2e–y isdx (A) ey = ex +

equation

equation

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4

Q.45

Q.46

The general solution of differential equation dy = sin3 x cos2 x + x ex isdx 1 1 (A) y = cos5 x + cosec3 x + (x + 1) ex + c 5 3 1 1 (B) y = cos5 x – cos3 x + (x – 1) ex + c 5 3 1 1 (C) y = – cos5 x – cos3 x – (x – 1) ex – c 5 3 (D) None of these The solution of the differential x(e2y – 1) dy + (x2 – 1) eydx = 0 is(A) ey + e–y = log x –

x2 +c 2

(B) ey – e–y = log x –

x2 +c 2

(C) ey + e–y = log x +

x2 +c 2

Q.50

Q.51

The

solution

(1 + x2)

the

differential

equation

of

the

differential

equation

dy = x (1 + y2) isdx

(A) 2 tan–1 y = log (1 + x2) + c (B) tan–1 y = log (1 + x2) + c (C) 2 tan–1 y + log (1 + x2) + c (D) None of these Q.52

Solution of the equation (ex + 1) y dy = (y + 1) ex dx is(A) c (y + 1) (ex + 1) + ey = 0 (B) c (y + 1) (ex – 1) + ey = 0 (C) c (y + 1) (ex – 1) – ey = 0 (D) c (y + 1) (ex + 1) = ey

Q.53

1 (B) tan–1 x – log (1 + x2) 2

(C) tan–1 x +

of

(A) ey = sin x + 1 (B) ey = cosec x + 1 (C) ey = cos x + 1 (D) ey = – sin x – 1

equation

The solution of the differential equation (1 + x2) (1+ y) dy + (1 + x) (1+ y2) dx = 0 is(A) tan–1 x + log (1 + x2) + tan–1 y + log (1 + y2) = c

+ tan–1 y –

solution

dy = e–y cos x, given that y (0) = 0 is– dx

(D) None of these Q.47

The

1 log (1 + y2) = c 2

The solution of the given differential equation dy + 2xy = y isdx (A) y = ce x x

2

(B) y = ce x

1 + log (1 + y2) = c 2 (D) None of these

Q.54

x

(D) y = ce x

(C) y = cex

1 log (1 + x2) + tan–1 y 2

2

2

d2y

= sec2 x + xex isdx 2 (A) y = log (sec x) + (x – 2) ex + c1x + c2 The solution of

(B) y = log (sec x) + (x + 2) ex + c1x + c2 Q.48

Q.49

The solution of y dx – x dy = 0 is(A) x = cy (C) x = c log x The

solution

dy + y = y2 isdx (A) y = 1 + cxy (B) y = log (cxy) (C) y + 1 = cxy (D) y = c + xy

x

the

differential

equation

(C) y = log (sec x) – (x + 2) ex + c1x + c2 (D) None of these

(B) xy = c (D) None of these of

differential

Q.55 equation

The equation of the curve which passes through 2y the point (1, 1) & whose slope is given by , is x (A) y = x2 (B) x2 – y2 = 0 (C) 2x2 + y2 = 3 (D) None of these

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5

Q.56

Q.57

Q.58

Equation of curve passing through (3, 9) which satisfies the differential equation dy 1 = x + 2 , isdx x (A) 6 xy = 3x2 – 6x + 29 (B) 6 xy = 3x3 – 29x + 6 (C) 6 xy = 3x3 + 29 x – 6 (D) None of these The equation of the curve through the point (1, 0) y 1 and whose slope is 2 isx x (A) (y – 1) (x + 1) + 2x = 0 (B) 2x (y – 1) + x + 1 = 0 (C) x (y – 1) (x + 1) + 2 = 0 (D) None of these dy = e–2y and y = 0 when x = 5, the value of dx x for y = 3 is(A) e5 (B) e6 + 1

The

e6 9 2

differential

equation

y

dy + dx

x

=

x y (C) log 1 tan = x + c 2 (D) None of these

Q.63

Question based on

a

differential

equation

The solution of the differential dy x = y (log y – log x + 1) isdx

equation

x (D) tan–1 = log x + c y

Differential Equation of the form dy/dx = f(ax + by + c)

dy The solution of the equation = (x + y)2 isdx (A) x + y + tan (x + c) = 0 (B) x – y + tan (x + c) = 0 (C) x + y – tan (x + c) = 0 (D) None of these

The solution of the differential dy = cot2 (x + y) isdx (A) y = x + 1/2 sin 2 (x + y) + c (B) y = x – 1/2 sin 2 (x + y) + c (C) y = x + 1/2 cos 2 (x + y) – c (D) None of these

The solution of the dy x2 = x2 + xy + y2 isdx

y (C) sin–1 = log x + c x

(A) y = xecx (C) y + ex = 0 Q.66

Q.61

Differential Equation of homogeneous type

y (B) tan–1 = – log x + c x

Q.65 Q.60

dy = (4x + y + 1)2 isdx (A) 4x – y + 1 = 2 tan (2x – 2c) (B) 4x – y – 1 = 2 tan (2x – 2c) (C) 4x + y + 1 = 2 tan (2x + 2c) (D) None of these

The solution of

y (A) tan–1 = log x + c x

(a is any constant) represents(A) a set of circles having centre on the y-axis (B) a set of circles centre on the x-axis (C) a set of ellipses (D) None of these Question based on

dy = sin (x + y) + cos (x + y) isdx

x y (B) log 1 tan = x + c 2

Q.64

(D) loge 6

The solution of

x y (A) log 1 tan + c = 0 2

If

(C)

Q.59

Q.62

(B) y + xecx = 0 (D) None of these

The solution of the equation

equation

dy x y = isdx x y

(A) c (x2 + y2)1/2 + e tan

1

(y / x)

(B) c (x2 + y2)1/2 = e tan

1

(y / x)

(C) c (x2 – y2) = e tan (D) None of these

1

(y / x)

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6

Q.67

Q.68

The solution of the (x2 + y2)dx = 2 xy dy is(A) x = c (x2 + y2) (B) x = c (x2 – y2) (C) x + c (x2 + y2) = 0 (D) None of these

differential

equation

Q.72

x

The solution of the equation x

dy = y – x tan dx

y isx

Q.73

(D) None of these Linear Differential Equation

(A) 4xy = x4 + c 1 (C) xy = x4 + c 4 Q.70

The

solution

of

the

differential

equation,

Q.74

(B) x3y =

x4 +c 4

The

solution

of

the

differential

equation

The solution of (1 + y2) dx = (tan–1 y – x) dy is (A) xe tan

1

y

= e tan

(B) xe tan

1

(B) xy = x4 + c

y

= (tan–1 y + 1) – c

(D) xy = 4x4 + c

(C) xe tan y = (tan–1 y – 1) + c (D) None of these

1

y

(tan–1 y – 1) + c

1

the

differential

equation

Q.75

dy + y = cos x isdx

1 (cos x + sin x) + ce– x 2 1 (B) y = (cos x – sin x) + ce– x 2 (C) y = cos x + sin x + ce–x (D) None of these

The solution of the differential equation dy x log x + y = 2 log x isdx

(A) y = log x + c (B) y = log x2 + c

The integrating factor of the differential equation dy + y = 2 log x isdx (A) log x (B) log (log x)

(x log x)

(A) y =

Q.71

x4 +c=0 4

dy + y cot x = 2 cos x isdx (A) y sin x + cos 2x = 2c (B) 2 y sin x + cos x = c (C) y sin x + cos x = c (D) 2y sin x + cos 2x = c

y (C) x sin = c x

The solution of dy y + = x2 isdx x

(A) x3y +

x4 =0 4 (D) None of these

(B) x sin y + c = 0

Q.69

dy + 3y = x isdx

(C) x3 y +

x (A) x sin + c = 0 y

Question based on

The solution of the equation

(C) ex Q.76

(D) x

The equation of the curve passing through the origin and satisfying the differential equation (1 + x2)

dy + 2xy = 4x2 isdx

(A) (1 + x2)y = x3 (B) 2 (1 + x2)y = 3x3 (C) 3 (1 + x2)y = 4x3 (D) None of these

(C) y log x = (log x)2 + c (D) y = x log x + c

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7

Q.77

The x

Q.78

solution

of

the

differential

equation

dy + y = x2 + 3x + 2 isdx

(A) xy =

3 x3 + x2 + 2x + c 3 2

(B) xy =

x4 + x3 + x2 + c 4

(C) xy =

x4 x3 + + x2 + c 4 3

(D) xy =

x4 + x3 + x2 + cx 4

The solution of the equation

dy 1 = isdx x y 1

(A) x = cey – y – 2 (B) y = x + cey – 2 (C) x + cey – y –2 = 0 (D) None of these Q.79

Integrating factor of the differential equation dy + y tan x – sec x = 0 isdx

(A) esin x (C) Question based on

Q.80

1 cos x

(B)

1 sin x

(D) ecos x

Equation reducible to linear form

The solution of differential equation dy + 1 = ex–y isdx

(A) ey = ex/ 2 + ce–x (B) ey = ex + ce–x (C) 3ey = ex/ 2 + ce–x (D) None of these

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8

LEVEL- 2 Q.1

The order of the differential equation whose general solution is given by y = (C1 + C2) cos

Q.7

(x + C3) – C4. e x C5 , where C1, C2,C3 , C4, C5 are arbitrary constants, is(A) 5 (B) 4 (C) 3 (D) 2 Q.2

Q.3

dy y 2 x 2 (A) = dx 2x y

(C) Q.4

Q.5

y y – cos2 , then the equation of the curve isx x e (A) y = tan–1 log x

The differential equation of all circles in the first quadrant which touch the coordinate axes is of order(A) 1 (B) 2 (C) 3 (D) None of these The differential equations of all circles passing through origin and having their centres on the x-axis is

d2y dx

2

=

y2 x 2 2 xy

x (B) y = x tan–1 log e e (C) y = x tan–1 log x (D) None of these

Q.8

The solution of the equation (1– x2) dy + xy dx = xy2 dx is(A) (y – 1)2 (1 – x2) = 0 (B) (y – 1)2 (1 – x2) = c2y2 (C) (y – 1)2 (1 + x2) = c2y2 (D) None of these

Q.9

The solution of

dy y 2 x 2 (B) = dx 2x

(D)

dy y 2 x 2 = dx 2 xy

The solution of the equation dy + y tan x = xm cos x isdx (A) (m+1) y = xm+1 cos x + c (m+1) cos x (B) my = (xm + c) cos x (C) y = (xm+1 + c) cos x (D) None of these

The slope of the tangent at (x, y) to a curve passing through is given by 1, 4

dy e x (sin 2 x sin 2 x ) = isdx y (2 log y 1)

(A) y2 (log y) – ex sin2 x + c = 0 (B) y2 (log y) – ex cos2 x + c = 0 (C) y2 (log y) + ex cos2 x + c = 0 (D) None of these Q.10

The solution of the differential equation (sin x + cos x) dy + (cos x – sin x)dx = 0 is(A) ex (sin x + cos x) + c = 0 (B) ey (sin x + cos x) = c (C) ey (cos x – sin x) = c (D) ey (sin x – cos x) = c

The solution of (cosec x log y)dy + (x2 y)dx = 0 islog y (A) + (2 – x2) cos x + 2 sin x = c 2 2

log y (B) + (2 – x2) cos x + 2x sin x = c 2 (log y) 2 + (2 – x2) cos x + 2x sin x = c 2 (D) None of these

(C) Q.6

The solution of the differential equation dy 3x 2 sin 2 x + y = is dx 1 x3 1 x3 1 (A) y (1 + x3) = x + sin 2x + c 2 1 (B) y (1 + x3) = cx + sin 2x 2 1 (C) y (1 + x3) = cx – sin 2x 2 x 1 (D) y (1 + x3) = – sin 2x + c 2 4

Q.11

The solution of the differential equation xy

dy (1 y 2 )(1 x x 2 ) = isdx (1 x 2 )

1 log (1 + y2) = log x – tan–1 x + c 2 1 (B) log (1 + y2) = log x + tan–1x + c 2 (C) log (1 + y2) = log x – tan–1x + c (D) log (1 + y2) = log x + tan–1 x + c

(A)

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9

Q.12

The solution of

Q.18

2

Solution of the differential equation sin

2

(x 1 y )dx + (y 1 x )dy = 0 is -

dy = a dx

with y (0) = 1 is(A) 1 x 2 + = c (B)

2

1 x – 1 y

(A) sin–1

2

x 1 y = a (C) sin 1 x

=c

(C) (1 + x2)3/2 + (1 + y2)3/2 = c (D) None of these Q.19 Q.13

Q.14

Q.15

The solution of the differential equation cos y log (sec x + tan x)dx = cos x log (sec y + tan y ) dy is(A) sec2 x + sec2 y = c (B) sec x + sec y = c (C) sec x – sec y = c (D) None of these

The solution of the differential equation dy = ex – y + x2e–y is dx (A) y = ex – y + x2e–y + c 1 (B) ey – ex = x3 + c 3

1 3 x +c 3 The solution of the equation

Q.16

Q.17

(C) y = log y + 1

(D) y = xy + c

The solution of dy xy = 2 isdx x y 2 2

(A) ay2 = e x / y (C) y = + + c

2

the

differential

(B) ay = ex/y (D) y = + y2 + c

6

d4y

dy + 8 + 5y = ex 4 dx dx

dy 3 (C) 1 dx

(D) y = x2

2/3

=4

d 3y dx 3

dy dy + 1 dx dx

2

Q.20

The general solution of the differential equation dy 1 cos 2 y + = 0, is given bydx 1 – cos 2x (A) tan y + cot x = c (B) tan y – cot x = c (C) tan x – cot y = c (D) tan x + cot y = c

Q.21

The solution of

dy + dx

1 – y2 1– x2

= 0, is

(A) tan–1 x + cot–1 x = c (B) sin–1 x + sin–1 y = c (C) sec–1 x + cosec–1 x = c (D) None of these

y = log y + c x

(B)

x y (D) sin =a x 1

4

dy y y = log 1 isdx x x y (A) log = cx x

y 1 = a

2 d3y dy (B) 5 3 + 8 1 + 5y = x8 dx dx

1 3 x +c 3

(D) ex – ey =

(B) sin

Which of the following differential equations has the same order and degree? (A)

The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is(A) x2 = y + 5 (B) y2 = x – 5 2 (C) y = x + 5 (D) x2 = y – 5

(C) ex + ey =

y 1 = a

Q.22 equation

dy + y tan x = sec x isdx (A) y sec x = tan x + c (B) y tan x = sec x + c (C) tan x = y tan x + c (D) x sec x = y tan y + c

The solution of

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LEVEL- 3 Q.1

The degree of the differential equation satisfying

Q.10

The equation of the curve whose subnormal is constant is(A) y = ax + b (B) y2 = 2ax + b 2 2 (C) ay – x = a (D) none of these

Q.11

The solution of

2

2

1 x + 1 y = a(x – y) is(A) 1 (B) 2 (C) 3 (D) none of these

Q.2

The differential equation of all non-horizontal lines in a plane is(A) (C)

Q.3

d2y dx 2

=0

dy =0 dx

(B) (D)

d2x dy 2

when(A) a = 0, b = 0 (B) a = 1, b = 2 (C) a = 0, b 0 (D) a = 2, b = 1

=0

dx =0 dy

The degree of the differential equation

Q.12

2

2 d2y 2 dy + = x sin d y is2 dx 2 dx dx (A) 1 (B) 2 (C) 3 (D) none of these

Q.4

Q.5

1 x 2 + 1 y 2 = (x 1 y 2 –y 1 x 2 )is(A) 1 (B) 2 (C) 3 (D) none of these

(C) y

The order of the differential equation of all circles of radius r, having centre on y-axis and passing through the origin is(A) 1 (B) 2 (C) 3 (D) 4

Q.7

Solution of differential equation (2x cos y + y2cos x) dx + (2y sin x – x2 sin y) dy = 0, is(A) x2 cos y + y2 sin x = c (B) x cos y – y sin x = c (C) x2 cos2 y + y2 sin2 x = c (D) None of the above The solution of the differential equation dy tan y tan y sin y – = , is dx x x2 x y (A) + n x = c (B) + n x = c sin y sin x (C) n y + x = c

Q.9

(A) y

The degree of the differential equation satisfying the relation

Q.6

Q.8

The differential equation of all parabolas having their axis of symmetry coinciding with the axis of X is-

(B) x

The order and degree the differential equation of all tangent lines to the parabola x2 = 4y is(A) 1, 2 (B) 2, 2 (C) 3, 1 (D) 4, 1

(D) n x + y = c

If ƒ(x), g(x) be twice differentiable functions on [0, 2] satisfying f ''(x) = g''(x), ƒ'(1) = 2g' (1) = 4 and ƒ(2) = 3 g(2) = 9, then ƒ(x) – g(x) at x = 4 equals(A) 0 (B) 10 (C) 8 (D) 2.

dy ax h = represents a parabola dx by k

d2y

2

d2x

2

dy + = 0 2 dx dx dy + = 0 2 dy dx d2y

dy + =0 dx 2 dx (D) none of these.

Q.13

Solution of the differential equation 2

x = 1 +xy

3

dy ( xy) 2 dy ( xy) 3 dy + + +.... dx 2! dx 3! dx

is– (A) y = loge(x) + C (B) y = (logex)2 + C (C) y = ± (log e x ) 2 2C (D) xy = xy + k Q.14

The solution of ydx – x dy + 3x2y2dx = 0 is3 3 x x (A) + e x = C (B) – e x = 0 y y (C) –

Q.15

3 x + ex = C y

(D) none of these

The solution of the differential equation xdy – ydx =

x 2 y 2 dx is-

(A) x +

x 2 y 2 = cx2

(B) y –

x 2 y 2 = cx

(C) x –

x 2 y 2 = cx

(D) y +

x 2 y 2 = cx2

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Q.16

The solution of the differential equation 1 dy (1 + y2) + x e tan y = 0, isdx

(A) xe 2 tan

1

y

= e tan

(B) (x – 2) = K e tan (C) 2x e tan (D) xe Q.17

Q.18

1

tan 1 y

y

1

1

y

1

y

+K

= tan–1y + K

Q.24

The degree of the differential equation y32/3 + 2 + 3y2 + y1 = 0 is(A) 1 (B) 2 (C) 3 (D) none of these The differential equation of system of concentric circles with centre (1, 2) isdy (A) (x – 2) + (y – 2) =0 dx dy (B) (x – 1) + (y – 2) =0 dx dy (C) (x + 1) + (y – 2) = 0 dx dy (D) (x + 2) + (y – 1) = 0 dx If sinx is an integrating factor of the differential equation dy/dx + Py = Q, then P can be(A) log sinx (B) cot x (C) sin x (D) log cos x

Q.20

The curve in which the slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is – (A) an ellipse (B) a parabola (C) a rectangular hyperbola (D) a circle

Q.22

y , then the value of cos is equal to 2 x 1 (A) x (B) (C) log x (D) ex x

y

= e 2 tan

The solution of the differential equation dy x sin x + (x cos x + sinx) y = sin x, dx When y(0) = 0 is (A) xy sin x = 1 – cos x (B) xy sin x + cos x = 0 (C) x sin x + y cos x = 0 (D) x sin x + y cos x = 1 Differential equation for y = A cos x + sin x, where A and B are arbitrary constants, is(A) (C)

d2y dx 2 d2y dx

2

– y = 0

(B)

– 2y = 0

(D)

y y If x sin dy = y sin – x dx and x x

y(1) =

K

Q.19

Q.21

Q.23

d2y dx 2 d2y dx 2

+ 2y = 0 + y = 0

The function f() = d d

dx

1 cos cos x satisfies 0

the differential equation (A) df + 2f() cot = 0 d df (B) –2f() cot = 0 d df (C) + 2f() = 0 d df (D) –2f() = 0 d

Statement type Questions Each of the questions given below consists of Statement-I (Assertion) and Statement-II (Reason). Use the following key to choose the appropriate answer. (A) If both Statement-I, Statement-II are true, and Statement-II is the correct explanation of Statement-I. (B) If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement-I (C) If Statement-I is true but Statement-II is false (D) If Statement-I is false but Statement-II is true. Q.25

Q.26

Statement-I: The differential equation whose solution is y = c1 e 2 x c 2 + c3 e x c4 , c1,c2, c3, c4 R is of order 4. Statement-II: Order of the differential equation is equal to the number of independent arbitrary constants in the solution of differential equation. Statement-I: y = c1 sin (x + c2) is a general solution of the differential equation

d2y dx 2

+y=0

Statement-II: y = asin x + bcos x is a trigonometric function. Q.27

Observe the following statements: Statement-I: Integrating factor of dy + y = x2 is ex dx Statement-II: Integrating factor of dy + P(x) y = Q(x) is dx

e

P ( x ) dx

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12

ANSWER KEY LEVEL- 1 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

B

A

C

A

C

B

A

A

A

D

A

B

D

C

C

A

B

C

A

Q.No. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

A

B

A

C

A

C

D

C

C

D

D

B

B

D

A

C

A

B

A

A

Q.No. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

C

A

A

C

B

A

C

A

A

A

A

D

A

A

A

C

A

C

B

C

Q.No. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

A

B

C

A

A

B

B

C

A

A

C

B

D

A

A

C

A

A

C

A

Ans. Ans. Ans.

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

Ans.

C

A

D

A

B

D

C

B

A

C

Q.No.

11

12

13

14

15

16

17

18

19

20

D

C

B

A

A

B

C

B

Ans.

B

A

Q.No.

21

22

Ans.

B

A

LEVEL- 3 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

B

D

A

A

A

A

A

B

B

C

A

C

A

D

C

B

B

B

C

Ques.

21

22

23

24

25

26

27

Ans.

B

B

C

A

D

B

C

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Practice Question Question based on

Q.1

LEVEL –1

Order and Degree of Differential Equation

(C) dy + dx = 0

(A) 1 Q.8

(B) 2

differential

(C) 3

equation

(D) 6

The order of the differential equation whose

(C) 1

(D) None of these

Q.9

The differential equation of all circles of radius a is of order(A) 2 (B) 3 (C) 4 (D) None of these

Q.10

The order of the differential equation of all circles

(B) 1, 1 (D) 2, 2

The order and degree of the differential equation 2

dy dy y=x + a 2 b 2 isdx dx

(A) 1, 2 (C) 1, 1

of radius r, having centre on y-axis and passing through the origin is(A) 1 (B) 2 (C) 3 (D) 4

(B) 2, 1 (D) 2, 2 Q.11

The order and degree of the differential equation dy 2 4 dx

2/3

=

4

– 4

(B) 3, 3 (D) 3, 2

d 3y

dx dx respectively(A) 4, 1 (C) 1, 1

3

+ 8

d2y dx

degree

d2y

dy +3 dx dx

are-

dx 2

The order and the degree of differential equation d4y

The 2

d2y

(A) 2, 2 (C) 2, 3

Q.6

the

solution is y = a cos x + b sin x + ce–x is(A) 3 (B) 2

The order and degree of differential equation

(A) 1, 2 (C) 2, 1

Q.5

of 3

(D) None of these

1 y 2 dx + y 1 x 2 dy = 0 are respectively-

Q.4

degree

2

d y dy (A) x – x + a = 0 (B) + xy = 0 dx 2 dx

Q.3

The

d2y dy + 1 = 0 is 2 dx dx

A differential equation of first order and first degree is2

Q.2

Q.7

Differential Equation

2

– 8

dy + 4y = 0 are dx

(B) 1, 4 (D) None of these

(A) 1 (C) 3 Q.12

of 2

the

differential

equation

d2y = x2 log 2 is dx

(B) 2 (D) None of these

The differential equation 2

4 d2y dy x 2 + + y = x2 is of dx dx

(A) degree 2 and order 2 (B) degree 1 and order 1 (C) degree 4 and order 3 (D) degree 4 and order 4

The order and degree of differential equation (xy2 + x) dx + (y – x2 y) dy = 0 are(A) 1, 2 (B) 2, 1 (C) 1, 1 (D) 2, 2

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1

Question based on

Q.13

Linear and Non-linear Differential Equation

Q.18

Which of the following equation is linear? (A)

dy 2 dy (A) y 1 = 2x dx dx

dy + xy2 = 1 dx

dy 2 dy (B) y 1 = 2x dx dx

dy (B) x2 + y = ex dx

(C)

dy + 3y = xy2 dx

(D) x

(C)

Which of the following equation is non- linear(A)

dy = cos x dx

(B)

Q.15

d2y 2

+y=0

Which of the following equation is linear ? 2

d2y (A) 2 + x2 dx

(B) y =

dx

2

(C) Q.20

dy =0 dx

dy dy + 1 dx dx

d y

2

dx 2

Q.17

dx 2

+ 9y = 0

d2y dx

2

d2y dx 2

–x

dy =0 dx

+x=0

(A) 2xy

dy + x2 = y2 dx

(B) 2xy

dy + x2 + y2 = 0 dx

dy + x2 = y2 dx

(D) None of these

(D)

d2y

Q.21

dx 2

The differential equation of all parabolas whose axes are parallel to y- axis is(A)

(C)

d 3y dx

3

d 3y dx

3

=0

+

Q.22

d2x dy

2

=0

(D)

d2x dy 2 d2y dx

2

=c

+2

dy =c dx

The differential equation of family of curve y = Aex + Be–x , where A and B are arbitrarily constants, is (A)

d2y dx

2

+y=0

2

dy + =0 dx

(B)

– 9y = 0

The differential equation of the family of curves represented by the equation x2 + y2 = a2 isdy dy (A) x + y =0 (B) y =x dx dx (C) y

(D)

dx

2

The differential equation of the family of curves

(C) xy

y = 4 sin 3x is a solution of the differential equationdy dy (A) + 8y = 0 (B) – 8y = 0 dx dx d2y

=0

d2y

represented by the equation (x – a)2 + y2 = a2 is-

Formation of Differential Equation

(C)

(B)

2

dy y + = log x dx x dy (D) y –4=x dx

Q.16

dy =0 dx

dy –x=0 dx

(C)

Question based on

+2

2

dy dy (D) + 3 +y=0 dx dx The differential equation of all the non-vertical lines in the xy- plane is-

(A)

dx dy 3 (D) x + = y2 dx dy / dx

(C) dx + dy = 0

d2y

3

dy + y2 = sin x dx

Q.19 Q.14

The differential equation of the family of curves y2 = 4a (x + a), where a is an arbitrary constant, is-

(D) None of these

(C) y

d2y dx 2

(B)

d2y dx 2

=y

2

dy – =0 dx

(D) None of these

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Q.23

Q.24

The differential equation for the line y = mx + c is (where c is arbitrary constant)(A)

dy =m dx

(B)

(C)

dy =0 dx

(D) None of these

Q.29

(D) y = ex cos x + c

Q.30

d2v

(B) y =

d2v

(C) y =

Q.31 Variable separable method

dy x = isdx y 2

(A) – = c (C) x2 + y2 = c

Q.27

Q.28

The solution of the differential equation

(A) y = tan (x2 + x + c)

2

y3

x2 log x – x2 + c 2

dy = (1 + x) (1 + y2) isdx

The general solution of the differential equation

x3

x2 +c 2

1 2 1 2 x + x log x + c 2 2 (D) None of these

2 dv (C) + =0 2 r dr dr (D) None of these

Q.26

dy = x log x isdx

(A) y = x2 log x –

2 dv (B) – =0 2 r dr dr

Q.25

The solution of

d2v

1 dv (A) + =0 2 r dr dr

Question based on

dy = ex (sin x + cos x) isdx

(A) y = ex (sin x – cos x) + c (B) y = ex (cos x – sin x) + c (C) y = ex sin x + c

dy +m=0 dx

The differential equation of the family of curves v A = + B, where A & B are arbitrary constants, isr

The solution of

(B) y = tan (2x2 + x + c) (C) y = tan (x2 – x + c) x3

y3

(B) + = c (D) x2 – y2 = c

The general solution of the equation (ey + 1) cos x dx + ey sin x dy = 0 is(A) (ey + 1) cos x = c (B) (ey – 1) sin x = c (C) (ey + 1) sin x = c (D) None of these The solution of the differential equation dy = sec2 x dx is(A) y = sec x tan x + c (B) y = 2 sec x + c 1 (C) y = tan x + c (D) None of these 2 The solution of the differential equation dy (1 + x2) = x isdx (A) y = tan–1 x + c (B) y = – tan–1 x + c 1 (C) y = loge (1 + x2) + c 2 1 (D) y = – log e(1 + x2) + c 2

x2 (D) y = tan x c 2

Q.32

The

solution

x sec y

of

the

differential

equation

dy = 1 isdx

(A) x sec y tan y = c (B) cx = sec y + tan y (C) cy = sec x tan x (D) cy = sec x + tan x Q.33

The solution of the equation dy 1 y2 + = 0 isdx 1 x 2

(A) x 1 y 2 – y 1 x 2 = c (B) x 1 y 2 + y 1 x 2 = c (C) x 1 y 2 + y 1 x 2 = c (D) None of these

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Q.34

Solution of the equation cos x cos y

Q.39

The solution of the differential (1 + cos x) dy = (1 – cos x) dx isx (A) y = 2 tan – x + c 2 (B) y = 2 tan x + x + c x (C) y = 2 tan + x + c 2 x (D) y = x – 2 tan + c 2

Q.40

If

Q.41

The solution of the differential equation sec2 x tan y dx + sec2 y tan x dy = 0 is(A) tan x = c tan y (B) tan x = c tan (x + y) (C) tan x = c cot y (D) tan x sec y = c

Q.42

The solution of

dy = – sin x sin y isdx

(A) sin y + cos x = c (B) sin y – cos x = c (C) sin y. cos x = c (D) sin y = c cos x Q.35

The

general

2

d y 2

=–

1 2

solution

of

the

equation

is-

dx x (A) y = log x + c1x + c2

(B) y = – log x + c1x + c2 1 + c1x + c2 x (D) None of these

(C) y = –

Q.36

Q.37

The general solution of the differential equation dy ey + (ey + 1) cot x = 0 isdx (A) (ey + 1) cos x = K (B) (ey + 1) cosec x = K (C) (ey + 1) sin x = K (D) None of these

x3 +c 3 (C) ey = ex + x3 + c

Q.38

x2 + log cos x + c 2

(B) y = ex + sin x +

x2 + log sec x + c 2

(C) y =

ex

x2 – sin x + + log cos x + c 2

(D) y = ex – sin x +

x2 + log sec x + c 2

= 0, then1 x 2 (A) y + sin–1 x = c (B) y2 + 2sin–1 x + c = 0 (C) x + sin–1 y = 0 (D) x2 + 2 sin–1 y = 1

dy 3e 2 x 3e 4 x = isdx e x e x

The general solution of differential equation dy (4 + 5 sin x) = cos x isdx 1 (A) y = log |4 + 5 sin x| + c 5 1 (B) y = log |4 + 5 cos x| + c 5 1 (C) y = – log |4 – 5 sec x| + c 5 (D) None of these

Q.44

The general solution of differential equation dy = log x is dx (A) y = x (log x + 1) + c (B) y + x (log x + 1) = c (C) y = x (log x – 1) + c (D) None of these

(D) None of these

(A) y = ex + sin x +

1

Q.43 (B) ey = ex + 2x + c

The solution of the differential dy = ex + cos x + x + tan x isdx

dy + dx

(A) y = e3x + c (B) –y = e3x – c (C) y = – e3x + c (D) None of these

The solution of the equation dy = e x–y + x2e–y isdx (A) ey = ex +

equation

equation

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Q.45

Q.46

The general solution of differential equation dy = sin3 x cos2 x + x ex isdx 1 1 (A) y = cos5 x + cosec3 x + (x + 1) ex + c 5 3 1 1 (B) y = cos5 x – cos3 x + (x – 1) ex + c 5 3 1 1 (C) y = – cos5 x – cos3 x – (x – 1) ex – c 5 3 (D) None of these The solution of the differential x(e2y – 1) dy + (x2 – 1) eydx = 0 is(A) ey + e–y = log x –

x2 +c 2

(B) ey – e–y = log x –

x2 +c 2

(C) ey + e–y = log x +

x2 +c 2

Q.50

Q.51

The

solution

(1 + x2)

the

differential

equation

of

the

differential

equation

dy = x (1 + y2) isdx

(A) 2 tan–1 y = log (1 + x2) + c (B) tan–1 y = log (1 + x2) + c (C) 2 tan–1 y + log (1 + x2) + c (D) None of these Q.52

Solution of the equation (ex + 1) y dy = (y + 1) ex dx is(A) c (y + 1) (ex + 1) + ey = 0 (B) c (y + 1) (ex – 1) + ey = 0 (C) c (y + 1) (ex – 1) – ey = 0 (D) c (y + 1) (ex + 1) = ey

Q.53

1 (B) tan–1 x – log (1 + x2) 2

(C) tan–1 x +

of

(A) ey = sin x + 1 (B) ey = cosec x + 1 (C) ey = cos x + 1 (D) ey = – sin x – 1

equation

The solution of the differential equation (1 + x2) (1+ y) dy + (1 + x) (1+ y2) dx = 0 is(A) tan–1 x + log (1 + x2) + tan–1 y + log (1 + y2) = c

+ tan–1 y –

solution

dy = e–y cos x, given that y (0) = 0 is– dx

(D) None of these Q.47

The

1 log (1 + y2) = c 2

The solution of the given differential equation dy + 2xy = y isdx (A) y = ce x x

2

(B) y = ce x

1 + log (1 + y2) = c 2 (D) None of these

Q.54

x

(D) y = ce x

(C) y = cex

1 log (1 + x2) + tan–1 y 2

2

2

d2y

= sec2 x + xex isdx 2 (A) y = log (sec x) + (x – 2) ex + c1x + c2 The solution of

(B) y = log (sec x) + (x + 2) ex + c1x + c2 Q.48

Q.49

The solution of y dx – x dy = 0 is(A) x = cy (C) x = c log x The

solution

dy + y = y2 isdx (A) y = 1 + cxy (B) y = log (cxy) (C) y + 1 = cxy (D) y = c + xy

x

the

differential

equation

(C) y = log (sec x) – (x + 2) ex + c1x + c2 (D) None of these

(B) xy = c (D) None of these of

differential

Q.55 equation

The equation of the curve which passes through 2y the point (1, 1) & whose slope is given by , is x (A) y = x2 (B) x2 – y2 = 0 (C) 2x2 + y2 = 3 (D) None of these

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Q.56

Q.57

Q.58

Equation of curve passing through (3, 9) which satisfies the differential equation dy 1 = x + 2 , isdx x (A) 6 xy = 3x2 – 6x + 29 (B) 6 xy = 3x3 – 29x + 6 (C) 6 xy = 3x3 + 29 x – 6 (D) None of these The equation of the curve through the point (1, 0) y 1 and whose slope is 2 isx x (A) (y – 1) (x + 1) + 2x = 0 (B) 2x (y – 1) + x + 1 = 0 (C) x (y – 1) (x + 1) + 2 = 0 (D) None of these dy = e–2y and y = 0 when x = 5, the value of dx x for y = 3 is(A) e5 (B) e6 + 1

The

e6 9 2

differential

equation

y

dy + dx

x

=

x y (C) log 1 tan = x + c 2 (D) None of these

Q.63

Question based on

a

differential

equation

The solution of the differential dy x = y (log y – log x + 1) isdx

equation

x (D) tan–1 = log x + c y

Differential Equation of the form dy/dx = f(ax + by + c)

dy The solution of the equation = (x + y)2 isdx (A) x + y + tan (x + c) = 0 (B) x – y + tan (x + c) = 0 (C) x + y – tan (x + c) = 0 (D) None of these

The solution of the differential dy = cot2 (x + y) isdx (A) y = x + 1/2 sin 2 (x + y) + c (B) y = x – 1/2 sin 2 (x + y) + c (C) y = x + 1/2 cos 2 (x + y) – c (D) None of these

The solution of the dy x2 = x2 + xy + y2 isdx

y (C) sin–1 = log x + c x

(A) y = xecx (C) y + ex = 0 Q.66

Q.61

Differential Equation of homogeneous type

y (B) tan–1 = – log x + c x

Q.65 Q.60

dy = (4x + y + 1)2 isdx (A) 4x – y + 1 = 2 tan (2x – 2c) (B) 4x – y – 1 = 2 tan (2x – 2c) (C) 4x + y + 1 = 2 tan (2x + 2c) (D) None of these

The solution of

y (A) tan–1 = log x + c x

(a is any constant) represents(A) a set of circles having centre on the y-axis (B) a set of circles centre on the x-axis (C) a set of ellipses (D) None of these Question based on

dy = sin (x + y) + cos (x + y) isdx

x y (B) log 1 tan = x + c 2

Q.64

(D) loge 6

The solution of

x y (A) log 1 tan + c = 0 2

If

(C)

Q.59

Q.62

(B) y + xecx = 0 (D) None of these

The solution of the equation

equation

dy x y = isdx x y

(A) c (x2 + y2)1/2 + e tan

1

(y / x)

(B) c (x2 + y2)1/2 = e tan

1

(y / x)

(C) c (x2 – y2) = e tan (D) None of these

1

(y / x)

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Q.67

Q.68

The solution of the (x2 + y2)dx = 2 xy dy is(A) x = c (x2 + y2) (B) x = c (x2 – y2) (C) x + c (x2 + y2) = 0 (D) None of these

differential

equation

Q.72

x

The solution of the equation x

dy = y – x tan dx

y isx

Q.73

(D) None of these Linear Differential Equation

(A) 4xy = x4 + c 1 (C) xy = x4 + c 4 Q.70

The

solution

of

the

differential

equation,

Q.74

(B) x3y =

x4 +c 4

The

solution

of

the

differential

equation

The solution of (1 + y2) dx = (tan–1 y – x) dy is (A) xe tan

1

y

= e tan

(B) xe tan

1

(B) xy = x4 + c

y

= (tan–1 y + 1) – c

(D) xy = 4x4 + c

(C) xe tan y = (tan–1 y – 1) + c (D) None of these

1

y

(tan–1 y – 1) + c

1

the

differential

equation

Q.75

dy + y = cos x isdx

1 (cos x + sin x) + ce– x 2 1 (B) y = (cos x – sin x) + ce– x 2 (C) y = cos x + sin x + ce–x (D) None of these

The solution of the differential equation dy x log x + y = 2 log x isdx

(A) y = log x + c (B) y = log x2 + c

The integrating factor of the differential equation dy + y = 2 log x isdx (A) log x (B) log (log x)

(x log x)

(A) y =

Q.71

x4 +c=0 4

dy + y cot x = 2 cos x isdx (A) y sin x + cos 2x = 2c (B) 2 y sin x + cos x = c (C) y sin x + cos x = c (D) 2y sin x + cos 2x = c

y (C) x sin = c x

The solution of dy y + = x2 isdx x

(A) x3y +

x4 =0 4 (D) None of these

(B) x sin y + c = 0

Q.69

dy + 3y = x isdx

(C) x3 y +

x (A) x sin + c = 0 y

Question based on

The solution of the equation

(C) ex Q.76

(D) x

The equation of the curve passing through the origin and satisfying the differential equation (1 + x2)

dy + 2xy = 4x2 isdx

(A) (1 + x2)y = x3 (B) 2 (1 + x2)y = 3x3 (C) 3 (1 + x2)y = 4x3 (D) None of these

(C) y log x = (log x)2 + c (D) y = x log x + c

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Q.77

The x

Q.78

solution

of

the

differential

equation

dy + y = x2 + 3x + 2 isdx

(A) xy =

3 x3 + x2 + 2x + c 3 2

(B) xy =

x4 + x3 + x2 + c 4

(C) xy =

x4 x3 + + x2 + c 4 3

(D) xy =

x4 + x3 + x2 + cx 4

The solution of the equation

dy 1 = isdx x y 1

(A) x = cey – y – 2 (B) y = x + cey – 2 (C) x + cey – y –2 = 0 (D) None of these Q.79

Integrating factor of the differential equation dy + y tan x – sec x = 0 isdx

(A) esin x (C) Question based on

Q.80

1 cos x

(B)

1 sin x

(D) ecos x

Equation reducible to linear form

The solution of differential equation dy + 1 = ex–y isdx

(A) ey = ex/ 2 + ce–x (B) ey = ex + ce–x (C) 3ey = ex/ 2 + ce–x (D) None of these

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8

LEVEL- 2 Q.1

The order of the differential equation whose general solution is given by y = (C1 + C2) cos

Q.7

(x + C3) – C4. e x C5 , where C1, C2,C3 , C4, C5 are arbitrary constants, is(A) 5 (B) 4 (C) 3 (D) 2 Q.2

Q.3

dy y 2 x 2 (A) = dx 2x y

(C) Q.4

Q.5

y y – cos2 , then the equation of the curve isx x e (A) y = tan–1 log x

The differential equation of all circles in the first quadrant which touch the coordinate axes is of order(A) 1 (B) 2 (C) 3 (D) None of these The differential equations of all circles passing through origin and having their centres on the x-axis is

d2y dx

2

=

y2 x 2 2 xy

x (B) y = x tan–1 log e e (C) y = x tan–1 log x (D) None of these

Q.8

The solution of the equation (1– x2) dy + xy dx = xy2 dx is(A) (y – 1)2 (1 – x2) = 0 (B) (y – 1)2 (1 – x2) = c2y2 (C) (y – 1)2 (1 + x2) = c2y2 (D) None of these

Q.9

The solution of

dy y 2 x 2 (B) = dx 2x

(D)

dy y 2 x 2 = dx 2 xy

The solution of the equation dy + y tan x = xm cos x isdx (A) (m+1) y = xm+1 cos x + c (m+1) cos x (B) my = (xm + c) cos x (C) y = (xm+1 + c) cos x (D) None of these

The slope of the tangent at (x, y) to a curve passing through is given by 1, 4

dy e x (sin 2 x sin 2 x ) = isdx y (2 log y 1)

(A) y2 (log y) – ex sin2 x + c = 0 (B) y2 (log y) – ex cos2 x + c = 0 (C) y2 (log y) + ex cos2 x + c = 0 (D) None of these Q.10

The solution of the differential equation (sin x + cos x) dy + (cos x – sin x)dx = 0 is(A) ex (sin x + cos x) + c = 0 (B) ey (sin x + cos x) = c (C) ey (cos x – sin x) = c (D) ey (sin x – cos x) = c

The solution of (cosec x log y)dy + (x2 y)dx = 0 islog y (A) + (2 – x2) cos x + 2 sin x = c 2 2

log y (B) + (2 – x2) cos x + 2x sin x = c 2 (log y) 2 + (2 – x2) cos x + 2x sin x = c 2 (D) None of these

(C) Q.6

The solution of the differential equation dy 3x 2 sin 2 x + y = is dx 1 x3 1 x3 1 (A) y (1 + x3) = x + sin 2x + c 2 1 (B) y (1 + x3) = cx + sin 2x 2 1 (C) y (1 + x3) = cx – sin 2x 2 x 1 (D) y (1 + x3) = – sin 2x + c 2 4

Q.11

The solution of the differential equation xy

dy (1 y 2 )(1 x x 2 ) = isdx (1 x 2 )

1 log (1 + y2) = log x – tan–1 x + c 2 1 (B) log (1 + y2) = log x + tan–1x + c 2 (C) log (1 + y2) = log x – tan–1x + c (D) log (1 + y2) = log x + tan–1 x + c

(A)

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Q.12

The solution of

Q.18

2

Solution of the differential equation sin

2

(x 1 y )dx + (y 1 x )dy = 0 is -

dy = a dx

with y (0) = 1 is(A) 1 x 2 + = c (B)

2

1 x – 1 y

(A) sin–1

2

x 1 y = a (C) sin 1 x

=c

(C) (1 + x2)3/2 + (1 + y2)3/2 = c (D) None of these Q.19 Q.13

Q.14

Q.15

The solution of the differential equation cos y log (sec x + tan x)dx = cos x log (sec y + tan y ) dy is(A) sec2 x + sec2 y = c (B) sec x + sec y = c (C) sec x – sec y = c (D) None of these

The solution of the differential equation dy = ex – y + x2e–y is dx (A) y = ex – y + x2e–y + c 1 (B) ey – ex = x3 + c 3

1 3 x +c 3 The solution of the equation

Q.16

Q.17

(C) y = log y + 1

(D) y = xy + c

The solution of dy xy = 2 isdx x y 2 2

(A) ay2 = e x / y (C) y = + + c

2

the

differential

(B) ay = ex/y (D) y = + y2 + c

6

d4y

dy + 8 + 5y = ex 4 dx dx

dy 3 (C) 1 dx

(D) y = x2

2/3

=4

d 3y dx 3

dy dy + 1 dx dx

2

Q.20

The general solution of the differential equation dy 1 cos 2 y + = 0, is given bydx 1 – cos 2x (A) tan y + cot x = c (B) tan y – cot x = c (C) tan x – cot y = c (D) tan x + cot y = c

Q.21

The solution of

dy + dx

1 – y2 1– x2

= 0, is

(A) tan–1 x + cot–1 x = c (B) sin–1 x + sin–1 y = c (C) sec–1 x + cosec–1 x = c (D) None of these

y = log y + c x

(B)

x y (D) sin =a x 1

4

dy y y = log 1 isdx x x y (A) log = cx x

y 1 = a

2 d3y dy (B) 5 3 + 8 1 + 5y = x8 dx dx

1 3 x +c 3

(D) ex – ey =

(B) sin

Which of the following differential equations has the same order and degree? (A)

The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is(A) x2 = y + 5 (B) y2 = x – 5 2 (C) y = x + 5 (D) x2 = y – 5

(C) ex + ey =

y 1 = a

Q.22 equation

dy + y tan x = sec x isdx (A) y sec x = tan x + c (B) y tan x = sec x + c (C) tan x = y tan x + c (D) x sec x = y tan y + c

The solution of

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10

LEVEL- 3 Q.1

The degree of the differential equation satisfying

Q.10

The equation of the curve whose subnormal is constant is(A) y = ax + b (B) y2 = 2ax + b 2 2 (C) ay – x = a (D) none of these

Q.11

The solution of

2

2

1 x + 1 y = a(x – y) is(A) 1 (B) 2 (C) 3 (D) none of these

Q.2

The differential equation of all non-horizontal lines in a plane is(A) (C)

Q.3

d2y dx 2

=0

dy =0 dx

(B) (D)

d2x dy 2

when(A) a = 0, b = 0 (B) a = 1, b = 2 (C) a = 0, b 0 (D) a = 2, b = 1

=0

dx =0 dy

The degree of the differential equation

Q.12

2

2 d2y 2 dy + = x sin d y is2 dx 2 dx dx (A) 1 (B) 2 (C) 3 (D) none of these

Q.4

Q.5

1 x 2 + 1 y 2 = (x 1 y 2 –y 1 x 2 )is(A) 1 (B) 2 (C) 3 (D) none of these

(C) y

The order of the differential equation of all circles of radius r, having centre on y-axis and passing through the origin is(A) 1 (B) 2 (C) 3 (D) 4

Q.7

Solution of differential equation (2x cos y + y2cos x) dx + (2y sin x – x2 sin y) dy = 0, is(A) x2 cos y + y2 sin x = c (B) x cos y – y sin x = c (C) x2 cos2 y + y2 sin2 x = c (D) None of the above The solution of the differential equation dy tan y tan y sin y – = , is dx x x2 x y (A) + n x = c (B) + n x = c sin y sin x (C) n y + x = c

Q.9

(A) y

The degree of the differential equation satisfying the relation

Q.6

Q.8

The differential equation of all parabolas having their axis of symmetry coinciding with the axis of X is-

(B) x

The order and degree the differential equation of all tangent lines to the parabola x2 = 4y is(A) 1, 2 (B) 2, 2 (C) 3, 1 (D) 4, 1

(D) n x + y = c

If ƒ(x), g(x) be twice differentiable functions on [0, 2] satisfying f ''(x) = g''(x), ƒ'(1) = 2g' (1) = 4 and ƒ(2) = 3 g(2) = 9, then ƒ(x) – g(x) at x = 4 equals(A) 0 (B) 10 (C) 8 (D) 2.

dy ax h = represents a parabola dx by k

d2y

2

d2x

2

dy + = 0 2 dx dx dy + = 0 2 dy dx d2y

dy + =0 dx 2 dx (D) none of these.

Q.13

Solution of the differential equation 2

x = 1 +xy

3

dy ( xy) 2 dy ( xy) 3 dy + + +.... dx 2! dx 3! dx

is– (A) y = loge(x) + C (B) y = (logex)2 + C (C) y = ± (log e x ) 2 2C (D) xy = xy + k Q.14

The solution of ydx – x dy + 3x2y2dx = 0 is3 3 x x (A) + e x = C (B) – e x = 0 y y (C) –

Q.15

3 x + ex = C y

(D) none of these

The solution of the differential equation xdy – ydx =

x 2 y 2 dx is-

(A) x +

x 2 y 2 = cx2

(B) y –

x 2 y 2 = cx

(C) x –

x 2 y 2 = cx

(D) y +

x 2 y 2 = cx2

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11

Q.16

The solution of the differential equation 1 dy (1 + y2) + x e tan y = 0, isdx

(A) xe 2 tan

1

y

= e tan

(B) (x – 2) = K e tan (C) 2x e tan (D) xe Q.17

Q.18

1

tan 1 y

y

1

1

y

1

y

+K

= tan–1y + K

Q.24

The degree of the differential equation y32/3 + 2 + 3y2 + y1 = 0 is(A) 1 (B) 2 (C) 3 (D) none of these The differential equation of system of concentric circles with centre (1, 2) isdy (A) (x – 2) + (y – 2) =0 dx dy (B) (x – 1) + (y – 2) =0 dx dy (C) (x + 1) + (y – 2) = 0 dx dy (D) (x + 2) + (y – 1) = 0 dx If sinx is an integrating factor of the differential equation dy/dx + Py = Q, then P can be(A) log sinx (B) cot x (C) sin x (D) log cos x

Q.20

The curve in which the slope of the tangent at any point equals the ratio of the abscissa to the ordinate of the point is – (A) an ellipse (B) a parabola (C) a rectangular hyperbola (D) a circle

Q.22

y , then the value of cos is equal to 2 x 1 (A) x (B) (C) log x (D) ex x

y

= e 2 tan

The solution of the differential equation dy x sin x + (x cos x + sinx) y = sin x, dx When y(0) = 0 is (A) xy sin x = 1 – cos x (B) xy sin x + cos x = 0 (C) x sin x + y cos x = 0 (D) x sin x + y cos x = 1 Differential equation for y = A cos x + sin x, where A and B are arbitrary constants, is(A) (C)

d2y dx 2 d2y dx

2

– y = 0

(B)

– 2y = 0

(D)

y y If x sin dy = y sin – x dx and x x

y(1) =

K

Q.19

Q.21

Q.23

d2y dx 2 d2y dx 2

+ 2y = 0 + y = 0

The function f() = d d

dx

1 cos cos x satisfies 0

the differential equation (A) df + 2f() cot = 0 d df (B) –2f() cot = 0 d df (C) + 2f() = 0 d df (D) –2f() = 0 d

Statement type Questions Each of the questions given below consists of Statement-I (Assertion) and Statement-II (Reason). Use the following key to choose the appropriate answer. (A) If both Statement-I, Statement-II are true, and Statement-II is the correct explanation of Statement-I. (B) If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement-I (C) If Statement-I is true but Statement-II is false (D) If Statement-I is false but Statement-II is true. Q.25

Q.26

Statement-I: The differential equation whose solution is y = c1 e 2 x c 2 + c3 e x c4 , c1,c2, c3, c4 R is of order 4. Statement-II: Order of the differential equation is equal to the number of independent arbitrary constants in the solution of differential equation. Statement-I: y = c1 sin (x + c2) is a general solution of the differential equation

d2y dx 2

+y=0

Statement-II: y = asin x + bcos x is a trigonometric function. Q.27

Observe the following statements: Statement-I: Integrating factor of dy + y = x2 is ex dx Statement-II: Integrating factor of dy + P(x) y = Q(x) is dx

e

P ( x ) dx

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12

ANSWER KEY LEVEL- 1 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

B

A

C

A

C

B

A

A

A

D

A

B

D

C

C

A

B

C

A

Q.No. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

A

B

A

C

A

C

D

C

C

D

D

B

B

D

A

C

A

B

A

A

Q.No. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

C

A

A

C

B

A

C

A

A

A

A

D

A

A

A

C

A

C

B

C

Q.No. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

A

B

C

A

A

B

B

C

A

A

C

B

D

A

A

C

A

A

C

A

Ans. Ans. Ans.

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

Ans.

C

A

D

A

B

D

C

B

A

C

Q.No.

11

12

13

14

15

16

17

18

19

20

D

C

B

A

A

B

C

B

Ans.

B

A

Q.No.

21

22

Ans.

B

A

LEVEL- 3 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

B

D

A

A

A

A

A

B

B

C

A

C

A

D

C

B

B

B

C

Ques.

21

22

23

24

25

26

27

Ans.

B

B

C

A

D

B

C

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13

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