Question

Solve: $\frac{dy}{dx}+2y\tan x=\sin x$.

Answer 1

$\frac{dy}{dx}+2ytanx=sinx$

This is in the form of $\frac{dy}{dx}+py=\theta$

where $p=2tanx,\theta =sinx$

$\therefore$ finding If $e^{\int pdx}$

$=e^{\int 2tanxdx}$

$=e^{2logsecx}$

$=e^{logse{c}^{2}x}$

$=se{c}^{2}x$

$y\left(If\right)=\int (Q\times IF)dx+c$ (c : const of integration)

$y\left(se{c}^{2}x\right)=\int sinxse{c}^{2}xdx+c$

$yse{c}^{2}x=\int \frac{sinx}{cosx}.\frac{1}{cosx}dx+c$

$=\int tanxsecxdx+c$

$=secx+c$

$\frac{d}{dx}\left(secx\right)$

$=tansecxdx$ vice -versa is also true

or $y=cosx+cco{s}^{2}x$