Question

Solve ${\displaystyle {(1+x)}^2\frac{d^{2}y}{{dx}^2}+(1+x)\frac{dy}{dx}+y=4\cos \left(\log (1+x)\right)}$

Answer 1

${\displaystyle {(1+x)}^2\frac{d^{2}y}{{dx}^2}+(1+x)\frac{dy}{dx}+y=4\cos \left(\log (1+x)\right)}$

Put x+1 = v ${\displaystyle \Rightarrow \frac{dv}{dx}=1}$

${\displaystyle \frac{dy}{dx}=\frac{dy}{dv}}$

The given eqn changes to ,

${\displaystyle v^2\frac{d^{2}y}{d{v}^2}+v\frac{dy}{dv}+y=4\cos \log v}$

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 ,

${\displaystyle \therefore y_c=c_1\cos z+c_2\sin z}$
For particular integral ,

${\displaystyle y_p=\frac{1}{f\left(D\right)}x=\frac{1}{D^2+1}4\cos z=4\frac{z}{2}\sin z=2z\sin z}$

${\displaystyle \therefore y_p=2z\sin z}$

The general solution of given diff. eqn is given by,

${\displaystyle y_g=y_c+y_p=c_1\cos z+c_2\sin z+2z\sin z}$
Resubstitute z and v,

${\displaystyle y_g=c_1\cos \left[\log (x+1)\right]+c_2\sin \left[\log (1+x)\right]+2\log (1+x)\sin \left[\log (1+x)\right]}$