The general solution of $\frac{d y}{d x}+y an x=2 x+x^2 an x$

Question

The general solution of $\frac{d y}{d x}+y 	an x=2 x+x^2 	an x$

Aicharya · Accepted Answer

Given differential equation,

$\because$ The differential equation is in linear form, so Integrating factor (I.F.) $=e^{\int \tan x d x}=\sec x$ So, solution of given differential Eq. (i), is $ \begin{aligned} y(\sec x) & =\int\left(2 x+x^2 \tan x\right) \sec x d x \\ & =\int 2 x \sec x d x+\int x^2 \tan x \sec x d x \\ & =\int 2 x \sec x d x+x^2 \sec x-\int 2 x \sec x d x \\ \Rightarrow \quad y \sec x & =x^2 \sec x+c \\ \Rightarrow \quad y & =x^2+c \cos x . \end{aligned} $

The general solution of $\frac{d y}{d x}+y \tan x=2 x+x^2 \tan x$

Solution