Villena Ecuaciones 2do Orden

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz



1

1.> @kunkbûcLbh`r`ckbnf`sl`s`gucld drl`ckdckd`hbkb`ct`skdcstnct`s. 1.>1.1   @kunkbdc`slbh`r`ckbnf`sl`drl`c sup`rbdr 1.; NcîfbsbsKunfbtntbvd

[`p`rsbgu`qu``f`stulbnct`9 • @cku`ctr` sdfukbdc`s g`c`rnf`s y/d pnrt pn rtbk bkuf ufnr nr`s `s l` l` @kun @kunkb kbdc dc`s `s Lb Lbh` h`r` r`ck ckbn bnf` f`s s l` s`guclddrl`c • L`t L`t`reb `rebc` c` @stnabf @stnabfbln blnl l lbcîebk lbcîebkn n kunctbt kunctbtntbv ntbvnn y/dkunfbtntbvne`ct`.

> ottp9 p9/ //sfbl`plh.kde de/ /r`n r`nl`r/ r/h huff/vbff`cn cn-`kunkbdc dc` `s-1ld-dr d-drl l`cpl cplh

>/>0

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz



1.> @KPNKBDC@[ LBH@Q@CKBNF @[ L@ [@GPCLD DQL@ DQ L@C CKD KDC C KD@HB KD@HBKB KB@C @C\@ \@[ [ KD KDC[ C[\N \NC\ C\@[ @[.. Pcn`kunkbûclbh`r`ck Pcn`kunkb ûclbh`r`ckbnfl`s`guclddrl` bnfl`s`guclddrl`c`sl`fnhdren9 c`sl`fnhdren9

y¼¼+p (x )y¼+ q (x)y

5 g ( x) 

  ) 5 =  s` ffnen @kunk [b g (x @kunkbûc bûc odedg odedgâc`n âc`n knsd knsd kdct kdctrn rnrb rbd: d: `s `s l`kb l`kbr, r, sb g (x  ) ≭ = s`ffnen @kunkbûccdodedgâc`n. Pcn `kun Pcn `kunkbû kbûc clbh lbh`r`c `r`ckbn kbnf fl`s`gucld l`s`gucld drl` drl`c ckdckd`hbk kdckd`hbkb`ct b`ct`skdcstn `skdcstnct`s ct`s `s l`fnhdren9

ny¼¼+  ay ¼+ ky  5 g ( x ) ldcl` n  , a   y k   ∏  BQ y n  ≭ = 

1.>.> .> @KPNKB NKBDC@[ @[ LBH@Q@CK @CKBNF@[ L@ [@ [@GP GPC CLD DQ DQL L@C KDC KD@HBKB@C\@[KDC[\NC\@[ODEDGÂC@N Pcn Pc n `ku `kunkb nkbûc ûc lb lbh`r h`r`ck `ckbn bnf f l` [` [`guc gucld ld Dr Drl`c l`c kdc kdc kd` kd`hb hbkb kb`ct `ct`s `s kdcst kdcstnct nct`s `s odedgâc`n`sl`fnhdren9

ny¼¼+ay    ¼+ky 5 =  Fnhuckbû Fnhuc kbûc" c" y ",sdf ",sdfukbû ukbûcg`c` cg`c`rnfl`f rnfl`fn`kun n`kunkbûclb kbûclbh`r`ckbn h`r`ckbnfnct` fnct`rbdr rbdr,`sl`f ,`sl`fn n (´]drqu quâ4).L â4).Ldcl` dcl`"" j "`sucnk "`sucnkdcstnct dcstnct`qu`l `qu`lnfng`c nfng`c`rn `rnfblnl fblnl hdren y ( x ) 5 j` (´]dr l`fnsdfukbûc. rx

@ctdck`s`fdai`tbvdnodr @ctdck`s `fdai`tbvdnodrns`rîonffnr ns`rîonffnr`fvnfdrl` `fvnfdrl`   r . Ab`c,l`fnsdfukbûcg`c`rnft`c`eds9

     ¼+ky Q``epfnzncld`c ny¼¼+ ay

y ‱ 5 jr` rx y ‱‱ 5 jr1 ` rx



5 = t`c`eds9

+ ajr` rx + kj` rx 5 = j` rx Znr1 + ar + k U 5  =

njr1 ` rx

 Nodrn ab`c, j ≭ = pdrqu` sb cd ttu uvbârneds fns sdfukbûc trbvbnf y kded rx 1 tneabâc ` ≭ = , `ctdck`s  nr + ar    + k 5 = . N `stn `xpr`sbûc s` fn l`cdebcn @kunkbûcNuxbfbnry`sùtbfpnrnonffnr  y`sùtbfpnrnonffnr    r . Das`rv`qu`fn`kunkbûcnuxbfbnr`sucn`kunkbûckunlrîtbknkuynsrnbk`s s`fnspu`l`l`t`rebcnr`epf s`fnspu`l`l`t`r ebcnr`epf`ncldfnhdreu `ncldfnhdreufng`c`rnf fng`c`rnf

1 ottp9 p9/ //sfbl`plh.kde de/ /r`n r`nl`r/ r/h huff/vbff`cn cn-`kunkbdc dc` `s-1ld-dr d-drl l`cpl cplh

1/>0

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz



r> , r1

∑a ¾

5

∑ (x) 5 j>`



rx >

y 1 (x) 5 j1 `

rx 1

 Fn Fnsd sdfu fukb kbûc ûcG`c G`c`r `rnf nf`s `stn tnrå rånln nlnln lnpd pdrf rfnk nkde deab abcn cnkb kbûc ûcfbc fbc`n `nfl` fl`fn fnss ssdf dfuk ukbd bdc` c`s s huclne`ctnf`s

 >` y (x)  5 j  

r>x

+  j1` rx  1

KnsdBB Lbskrbebc Lbskrb ebcnct` nct` k`r k`rd d a 1 ∑   y r1 sdc rnåk`s r`nf`s ` bgunf`s. @c`st`knsdfnsdfukbûcG`c`r @c`st`knsdfnsdf ukbûcG`c`rnfs`rån9 nfs`rån9 y (x )  5 j  > ` rx + j1x` rx 

KnsdBBB

 + °b  y r1 Lbskrbebcnct` Lbskrbebcnc t` c`gntbvd a 1 ∑   5   κ rnåk`skdepf`inskdciugnlns

 5 κ ∑ °b sdc

Q``epfnzncld `c y (x)  5 K  >` rx + K1 ` rx t`c`eds9   >

1

y (x) 5K>` (κ +°b )x

+ K1 ` (κ∑°b )x x bx x bx y (x) 5K>` κ ` ° + K1 ` κ ` ∑° y (x) 5 ` κx Z K>` °bx + K1 ` ∑°bx U



  b x

b   x

  °x  5 kds  °x ∑ b s`c

  °x  y ` ∑ ° 5 kds  °x + b s`c Q``epfnzncldt`c`eds9

Kded ` °



κx

ZK>(kds °x + b s`c °x) + K1 (kds °x ∑ b s`c °x)U κ y (x) 5 ` x Z(K>+K1 ) kds °x + (K>b + K1b ) s`c °x U y (x) 5 `

  `  κx Zj>  s`c( °x ) + j1 kds(°x )U  ]drfdtnctdfnsdfukbûcs`rån y (x  ) 5

@cku`ctr`fnsdfukbûcg`c`rnfpnrn y ‱‱ ∑  1 y 5 =  [DFPKBÛC9 1

@c`st`knsdfn`kunkbûcnuxbfbnrs`rån r



∑  1 5 =

 

ottp9 p9/ //sfbl`plh.kde de/ /r`n r`nl`r/ r/h huff/vbff`cn cn-`kunkbdc dc` `s-1ld-dr d-drl l`cpl cplh

; ;/>0

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz

 (r ∑ 8)(r + 1) 5 =

Onffncldfnsrnåk`st`c`eds

r 5 8

r 5 ∑1



]drtnctd9

y> (x) 5 j>`

8x

∑1x y 1 (x)

5

j1 ` 8x

y (x) 5 j `

+ j ` ∑1x

>

1

]dl`edskdeprdanr  qu``h`ktbvne qu``h`ktbvne`ct``stn`sfnhuckbûcqu`sntbshnk` `ct``stn`sfnhuckbûcqu`sntbshnk`fn`kunkbûclbh`r`c fn`kunkbûclbh`r`ckbnflnln. kbnflnln. Dat`cgnedsfnprbe`rnyfns`guclnl`rbvnln

y ‱ 5 8j>`

8x

∑ 1j1 ` ∑1x + ` 8x Fu`gd,r``epfnzncld

;8j>` 8x

  j ` 8  x +  ?j ` ∑1x  + 1j>` 8x ∑ >1j1 ` ∑1 x 5 = > 1

=5=



@cku`ctr`fnsdfukbûcg`c`rnfpnrn 1y‱‱  ∑  ;y  ‱ + y 5 = , y(=) 5  >     y ‱(=) 5 >  [DFPKBÛC9  1

@c`st`knsdfn`kunkbûcnuxbfbnrs`rån 1r

∑ ;  r + > 5 =

Onffncldfnsrnåk`st`c`eds

r 5

; ¾ 2 ∑ ) <

r 5 ; ¾ > < r>

5  > 

r1



5

> 1   x

]drtnctd,fn sdfukbûcg`c`rnf s`rån9

> 1

 

y(x)  5 j>`

+ j1 `

x



Kdedfnskd dfnskdclb clbkbdc`sbc kbdc`sbcbkbnf`s`stî bkbnf`s`stîcln clnlns lnsl`a l`a`e `eds`c ds`ckdc kdctrnr trnrfnskdcs fnskdcstnct tnct`s `s j> y j1  

x

+ j1 `

>

=

+ j1 `

>

y(x) 5 j>` Kded y (=  ) 5 >

`ctdck`s

y(=) 5 j>`

1

1

x

=



> 5 j> + j1 Dat`cb`cldfnprbe`rnl`rbvnln9 x

 >` y ‱(x)  5 j

 

+

> j1 ` 1

> 1

x



< ottp9 p9/ //sfbl`plh.kde de/ /r`n r`nl`r/ r/h huff/vbff`cn cn-`kunkbdc dc` `s-1ld-dr d-drl l`cpl cplh

0

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz



 

x

y ‱(x) 5 j>` Kded y ‱(=    ) 5 > `ctdck`s y ‱(=) 5 j>`

=

> 5 j> +

>

+

1 >

+ >

1

j1 `

 

1

x

j1 ` =  >

1

j1

1

⎧ > 5 j + j > 1 Q`sdfvb`cld fvb`cldsbeu sb euftftîc`ne îc`ne`ct` ⎩ ⎯ > ⎩⎢ > 5 j> + 1 j1 ]drtnctd,fn sdfukbûcpnrtbkufnr `s9 y( x )

>

 j1  5 =  y j>  5 > 

t`c`eds9 t`c`eds9

5 `x 

@cku`ctr`fnsdfukbûcg`c`rnfpnrn y ‱‱ + `

+ j1x` ∑1x 

@cku`ctr`fnsdfukbûcg`c`rnfpnrn y ‱ +  8y ‱  + >; y 5 = : y(=) 5  > :   y ‱(=) 5 > [DFPKBÛC9 @c`st`knsdfn`kunkbûcnuxbfbnrs`rån

5

r> , r1

5 ∑ 8 ¾ ∑ >8 5

r> , r1

5

r>

1 1

Onffncldfnsrnåk`st`c`eds9 r> , r1

@c`st`knsd

∑ 8 ¾ ;8 ∑ )(>;)

r> , r1

∑> 5 b

∑ 8 ¾ >8 ∑ >



1 ∑ 8 ¾   s`c(1x) + j1 kds(1x)U  y (x) 5 `

Kded y(=  )  5 > `ctdck`s

y (=) 5 ` ∑;(=) Zj> s`c(1(=)) + j1 kds(1(=))U

> 5 (>)Zj>(=) + j1 (>) U



> 5 j1

0 ottp9 p9/ //sfbl`plh.kde de/ /r`n r`nl`r/ r/h huff/vbff`cn cn-`kunkbdc dc` `s-1ld-dr d-drl l`cpl cplh

0/>0

 

0/10/1=>?

Xbff`c n-@kunkbd c `s1ld Drl`c.plh-sfbl`plh.kde

EdbsâsXbff`cnEuþdz



∑;x Z1 j kds(1x) ∑ 1j s`c(1x )U∑ ;`∑;x Zj s`c(1x) + j kds(1x)U > 1 > 1 Kdedy ‱(=  ) 5 > `ctdck`s y‱(=) 5 `∑;(=) Z1 j kds(=) ∑ 1j s`c( =)U∑ ;`∑;(=) Zj s`c( =) + j kds( =)U  y‱(x) 5 `

> > 5 1j> ∑ ;j1

> 1

+

>

1

1

; j15 j> 1

Q`sdfvb`cldsbeuftîc`ne`ct` > + ; (>) 5 j>  1

1

j> 5 1  

   

]drtnctd,fnsdfukbûcg`c`rnfs`rån y(x) 5 `

∑; x Z1 s`c(1x) + kds(1x)U 

@cku`ctr`fnsdfukbûcl`fnssbgub`ct`s`kunkbdc`slbh`r`ckbnf`sl`s`guclddrl`c >. y ‱‱    y 

: y (=) 5  > ,    y¼(=) 5 > 

∑   y‱‱  ∑  y  5 =  y‱‱  ∑  y ¼5 = : y (=) 5  > ,    y¼(=) 5 > 

1. ;.  ,    y¼(=) 5 > 

+  y ¼55 ==  y‱‱  + > 1

y‱  + 1 y 

5 =

>=. y ‱‱ ∑  8y ‱  + 2 y

5=

1.>.>.>NCÎFB[B[L@@[\NABFBLNLLBCÎEBKN @c `f @c `f kn knpå påtu tufd fd nc nct` t`rb rbdr dr s` s` e` e`ck ckbd bdcû cû qu qu` ` fn fn `s `stn tnab abfb fbln lnl l lbcî lbcîeb ebkn kn l` l` uc ucn n    (t) . trny`ktdrbn y (t) s`fnl`t`re  bcnk kd dc fåe y t↑ ∓

]dl`edsbrncnfbzncldpdrknsds.  

  r>t

r1 t

>

1

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KnsdBB, y (t) 5 j ` + j t` .[bfnsrnåk`s .[bfnsrnåk`s sdcr`nf sdcr`nf`s` `s`bgun bgunf`s f`s `ctd `ctdck`s ck`s   r  tb`c`qu`s`r  c`gntbvn( r 6 = ) pn pnrn rnqu qu` `fn fntr trny ny`k `ktd tdrb rbn ns` s`n nlb lbcî cîeb ebkn kne` e`ct ct` ``s `stn tnaf af` `  

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Knsd BBB y (t) 5  ` κ Zj kds   ut+ j s`c utU  [b fns rnåk`s sdc kdepf`ins kdciugnlns`ctdck`sfn pnrt`r`nf   κ tb`c`qu`s`rc`gntbvn  ( κ 6 = ) p pnrn q qu` ffn  

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L`t`rebc`kunfbtntbvne`ct`fn`stnabfblnllbcîebknpnrn y ‱‱‱ ∑ >=  y"+ 17y '∑>? y 5 ;  [DFPKBÛC9 @epf`ncld`f\`dr`enl`Qduto.Fn`kunkbûcnuxbfbnr`s r; ∑ >=  r  1 +  17 r ∑ >? 5 =

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