Question

A solution of differential equation $\left(se{c}^{2}y\right)\frac{dy}{dx}+2xtany=x^3$, is

• A. $2tany=c.e^{-x^2}+x^2-1$
• B. $tany=c.e^{-x^2}+x^2-1$
• C. $tany=c{e}^{x^2}+x^2-1$
• D. None of these

Answer 1

The correct option is A

$2tany=c.e^{-x^2}+x^2-1$

$tany=z\Rightarrow se{c}^{2}y\frac{dy}{dx}=\frac{dz}{dx}\frac{dz}{dx}+2xz=x^3$
I.F=$e^{\int 2xdx}=e^{x^2}$
Solution is $z.e^{x^2}=\int x^3.e^{x^2}dx$
Let $t=x^2\Rightarrow dt=2xdx\Rightarrow tany.e^{x^2}=\frac{1}{2}[t.e^t-e^t]+c\Rightarrow 2tany=c{e}^{-x^2}+x^2-1$