Let y=y(x) be the solution of the differential equation x(dy)/(dx)-2y=2+3 x, x (-( )/(2),(…

Question

Let $y=y(x)$ be the solution of the differential equation $\sec x\frac{dy}{dx}-2y=2+3\sin x$, $x\in (-\frac{\pi }{2},\frac{\pi }{2})$,
$y(0)=-\frac{7}{4}$. Then $y(\frac{\pi }{6})$ is equal to :

Accepted Answer

$\begin{gathered} \frac{dy}{dx}-2\cos x\cdot y=\cos x(2+3\sin x) \ I\cdot F={e}^{-2\sin x} \ G\cdot Sy\cdot {e}^{-2\sin x}=\int {e}^{-2\sin x}\cos x(2+3\sin x)dx \ \begin{gathered} \sin x=t \ \cos x\cdot dx=dt \end{gathered} \end{gathered}$

$y={e}^{-2\sin x}(\frac{2+3\sin x}{-2}-\frac{3}{4})+c$

When $x=0,c=0$

When $x=\frac{\pi }{6},y=-\frac{5}{2}$

Solutions · 1