# Solve the Following Differential Equations: ( 1 + X 2 ) D Y D X − 2 X Y = ( X 2 + 2 ) ( X 2 + 1 ) - Mathematics

Sum

Solve the following differential equation:-
$\left( 1 + x^2 \right)\frac{dy}{dx} - 2xy = \left( x^2 + 2 \right)\left( x^2 + 1 \right)$

#### Solution

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
$I.F.= e^{\int - \frac{2x}{1 + x^2}dx} = \frac{1}{1 + x^2}$

$y\left( \frac{1}{1 + x^2} \right) = \int\left( \frac{x^2 + 2}{x^2 + 1} \right)dx$
$\Rightarrow y\left( \frac{1}{1 + x^2} \right) = \int\left( 1 + \frac{1}{1 + x^2} \right)dx$
$\Rightarrow y\left( \frac{1}{1 + x^2} \right) = x + \tan^{- 1} x + C$
$\Rightarrow y = \left( x + \tan^{- 1} x + C \right)\left( 1 + x^2 \right)$
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 22 Differential Equations
Exercise 22.1 | Q 29 | Page 106