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Question
Solve `(1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))`
Solution
`(1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))`
Put x+1 = v `=> (dv)/(dx)=1`
`(dy)/(dx)=(dy)/(dv)`
The given eqn changes to ,
`v^2(d^2y)/(dv^2)+v(dy)/(dv)+y=4coslogv`
Now put log v = z ∴v=๐๐
[๐ซ(๐ซ−๐)+๐ซ+๐]๐=๐๐๐๐ ๐
∴ (๐ซ๐+๐)๐=๐๐๐๐ ๐
For complementary solution ,
๐(๐ซ)=๐
∴ (๐ซ๐+๐)=๐
Roots are : i,-i
The complementary solution of given diff. eqn is ,
`therefore y_c=c_1cosz+c_2sinz`
For particular integral ,
`y_p=1/(f(D))x=1/(D^2+1)4cosz=4z/2sinz=2zsinz`
`therefore y_p=2zsinz`
The general solution of given diff. eqn is given by,
`y_g=y_c+y_p=c_1cosz+c_2sinz+2zsinz`
Resubstitute z and v,
`y_g=c_1cos[log(x+1)]+c_2sin[log(1+x)]+2log(1+x)sin[log(1+x)]`
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