Question

Subject differential equationDay. Month.Year. ..(y-secx) dx + tanxdy...o​

Answer 1

$(y-\sec x)\,\mathrm dx+\tan x\,\mathrm dy=0$

Divide both side by $\mathrm dx$ and rearrange terms to get a linear ODE;

$\tan x\dfrac{\mathrm dy}{\mathrm dx}+y=\sec x$

Multiply both sides by $\cos x$:

$\sin x\dfrac{\mathrm dy}{\mathrm dx}+\cos x\,y=1$

The left side can be condensed as the derivative of a product:

$\dfrac{\mathrm d}{\mathrm dx}(\sin x\,y)=1$

Integrate both sides, then solve for $y$:

$\sin x\,y=x+C\implies\boxed{y(x)=x\csc x+C\csc x}$