9. differential equations — ગણિત ધોરણ 12 સાયન્સ — Question

$\frac{{dy}}{{dx}} + y\,f(x) = g(x)$

$I.F.$$ = {e^{\int {f(x)dx} }} = {e^{\int {2\tan x\,dx} }} = {e^{2\log (\sec x)}} = {e^{\log {{\sec }^2}x}} = {\sec ^2}x$

Hence, the solution is $y\,({\rm{I}}{\rm{.F}}{\rm{.)}} = \int {g(x)\,{\rm{I}}{\rm{.F}}{\rm{.}}\,dx + c} $

$y({\sec ^2}x) = \int {\sin x\,{{\sec }^2}x\,dx + c} $

==> $y{\sec ^2}x = \int {\sec x\,\tan x\,dx + c} $ ==> $y{\sec ^2}x = \sec x + c$.