Question

Solve the following differential equations:

$\left(1+x^2\right)\frac{dy}{dx}-2xy=\left(x^2+2\right)\left(x^2+1\right)$ [CBSE 2005]

Answer 1

Given, $\left(1+x^2\right)\frac{dy}{dx}-2xy=\left(x^2+2\right)\left(x^2+1\right)$
$\Rightarrow \frac{dy}{dx}-\frac{2x}{\left(1+x^2\right)}y=\left(x^2+2\right)$
This is a linear differential equation.
I.F.$=e^{\int -\frac{2x}{1+x^2}dx}=\frac{1}{1+x^2}$
$$\begin{gathered} y\left(\frac{1}{1+x^2}\right)=\int \left(\frac{x^2+2}{x^2+1}\right)dx \\ \Rightarrow y\left(\frac{1}{1+x^2}\right)=\int \left(1+\frac{1}{1+x^2}\right)dx \\ \Rightarrow y\left(\frac{1}{1+x^2}\right)=x+{\tan }^{-1}x+C \\ \Rightarrow y=\left(x+{\tan }^{-1}x+C\right)\left(1+x^2\right) \end{gathered}$$