Question

Solve `(1+x)^2(d^2y)/(dx^2)+(1+x)(dy)/(dx)+y=4cos(log(1+x))`

Answer 1

`(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=𝒆^𝒛

[𝑫(𝑫−𝟏)+𝑫+𝟏]𝒚=𝟒𝒄𝒐𝒔 𝒛
∴ (𝑫^𝟐+𝟏)𝒚=𝟒𝒄𝒐𝒔 𝒛
For complementary solution ,

𝒇(𝑫)=𝟎
∴ (𝑫^𝟐+𝟏)=𝟎
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)]`