Question

Solve the differential equation $\frac{dy}{dx}+y\sec x=\tan x,0\le x<\frac{\pi }{2}$.

Answer 1

Given $\frac{dy}{dx}+y\sec x=\tan x$

It is of the form $\frac{dy}{dx}+yp\left(x\right)=q\left(x\right)$

Integrating factor $e^{\int p\left(x\right)}$

Integrating factor $e^{\int \sec x}$

$=e^{\ln |\sec x+\tan x|}=|\sec x+\tan x|$

Solution: $y|\sec x+\tan x|=\int \tan x(\sec x+\tan x|)dx$

$=\int \tan x\sec xdx+\int {\tan }^{2}xdx$

${\tan }^{2}x={\sec }^{2}x-1$

$=\sec x+\tan x-x+c$

therefore final solution is: $y|\sec x+\tan x|=\sec x+\tan x-x+c$