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A Differential Equation is a mathematical equation that relates some functions with its derivatives. Differential equations play a prominent role in many disciplines including engineering, physics, economics and biology. In biology and economics, differential equations are used to model the behaviour of complex systems. Many fundamental laws of physics and chemistry can be formulated as differential equations. In mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions, the set of functions that satisfy the equation. Differential Equation: An equation involving derivatives of one or more dependent variable with respect to one or more independent variable is called a Differential Equation. For example, 𝑑𝑦 𝑑𝑥

= 𝜆𝑦

is differential equation where λ is a constant, 𝑥 is an independent variable and 𝑦 is a dependent variable.

Differential Equation are of two types 1. Ordinary differential equation 2. Partial differential equation Ordinary differential equation: A differential equation involving derivatives with respect to a single independent variable is called an ordinary differential equation. For example,

𝑑𝑦 𝑑𝑥

= 2 sin 𝑥 , 𝑥 ∈ (0, 𝑎], 𝑎 > 0

is an ordinary differential equation where 𝑥 is an independent variable and 𝑦 is a dependent variable. Partial differential equation: A differential equation involving derivatives with respect to more than one independent variable is called an partial differential equation. For example, 𝜕2 𝑢 𝜕𝑥 2

𝜕2 𝑢

+ 𝜕𝑦 2 = 0 where 𝑢 = 𝑢(𝑥, 𝑦) and (𝑥, 𝑦) 𝜖 Ω = (0,1) × (0,1)

is a partial differential equation where the domain of definition Ω is the open unit rectangle of dimension 2.

Order: The order of a differential equation is the order of the highest order derivative occurring in it. For example, 𝑑2𝑦 𝑑𝑦 3 + 5 ( ) − 4𝑦 = 𝑒 𝑥 𝑑𝑥 2 𝑑𝑥 is the second order ordinary differential equation.

Degree: The degree of a differential equation is the degree of the highest order derivative which occurs in it, after the differential equation has been made free from radicals and fractions as far as the derivatives are concerned.

For example, 3

𝑑3𝑦 𝑑2𝑦 𝑑𝑦 5 + ( 2) + ( ) + 𝑦 = 7 𝑑𝑥 3 𝑑𝑥 𝑑𝑥 is the second order differential equation of degree one.

Linear differential equation: A linear differential equation is that no products of the function 𝑦(𝑥) and its derivatives and neither the function nor its derivative occur to any power other than first power. A general form of linear 𝑛𝑡ℎ order differential equation is 𝑎𝑛 (𝑥)𝑦 𝑛 + 𝑎𝑛−1 (𝑥)𝑦 𝑛−1 + ⋯ + 𝑎1 (𝑥)𝑦 = 𝑔(𝑥) ……………..(1) For example, 𝑑𝑦 𝑑𝑥

+ 𝑥𝑦 = 𝑒 𝑥 , 𝑥 ∈ (0, 𝑎], 𝑎 > 0 is linear in y.

If a differential equation cannot be written in the form (1) then it is called a non-linear differential equation. For example, 𝑑𝑦

𝑦 𝑑𝑥 + 𝑥 = sin 𝑥 , 𝑥 ∈ (0, 𝑎], 𝑎 > 0 is a non-linear differential equation. Homogeneous differential equation: When 𝑔(𝑥) = 0, (1) becomes, 𝑎𝑛 (𝑥)𝑦 𝑛 + 𝑎𝑛−1 (𝑥)𝑦 𝑛−1 + ⋯ + 𝑎1 (𝑥)𝑦 = 0 This equation is called homogeneous. If 𝑔(𝑥) ≠ 0, then the equation is called non- homogeneous.

For example, 𝑦 ′ + 𝑦 = 0 is a homogeneous equation. 𝑦 ′ + 2𝑦 = 𝑥 is a non- homogeneous equation. Initial value problem: An initial value problem is an ordinary differential equation together with a specified value (initial condition) of the unknown function at the initial point in the domain of the solution. On some interval I containing the point 𝑥0 , 𝑑𝑛 𝑦 𝑑𝑥 𝑛

= 𝑓(𝑥, 𝑦, 𝑦 ′ , … 𝑦 𝑛−1 )

and 𝑦(𝑥0 ) = 𝑦0 , 𝑦 ′ (𝑥0 ) = 𝑦1 , … 𝑦 𝑛−1

………………(2)

where 𝑦0 , 𝑦1 , … 𝑦𝑛−1 are arbitrarily specified real constants. The values of 𝑦(𝑥) and its first 𝑛 − 1 derivatives at a single point 𝑥0 ,𝑦(𝑥0 ) = 𝑦0 , 𝑦 ′ (𝑥0 ) = 𝑦1 , … 𝑦 𝑛−1 (𝑥0 ) = 𝑦𝑛−1 are called initial conditions. For example, 𝑦 ′ = 𝑥, 𝑥 𝜖 (0, 𝑋], 𝑋 > 0 with 𝑦(0) = 1 is the first order initial value problem. Boundary value problem: A boundary value problem is a differential equation together with a set of additional constraints prescribed at more than one point, called the boundary conditions. There are four important kinds of boundary conditions. They are 1. Dirichlet or 1𝑠𝑡 kind 2. Neumann or 2𝑛𝑑 kind 3. Robin or 3𝑟𝑑 kind or Mixed kind Dirichlet condition:

The specification of the unknown function at the boundaries of the domain of the independent variable is known as a dirichlet boundary condition. Neumann condition: If the derivative is specified, then this is known as a neumann boundary condition. Mixed condition: When the boundary condition is an equation that involves both the value of the function and the value of its derivative, it is known as a mixed boundary condition. Example: Consider the 2𝑛𝑑 order linear differential equation 𝑦" + 𝑝(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 𝑟(𝑥),

𝑎