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

Question

Let $y=y(x)$ be the solution of the differential equation
$$\sec x\frac{dy}{dx} - 2y = 2+3\sin x,\quad x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),\qquad y(0)=-\frac74.$$
Then $y\!\left(\frac{\pi}{6}\right)$ is equal to:

Accepted Answer

$$\frac{dy}{dx} - 2\cos x\cdot y = \cos x(2+3\sin x).$$

I.F. $=e^{-2\sin x}$.
$$y\cdot e^{-2\sin x} = \int e^{-2\sin x}\cos x(2+3\sin x)\,dx.$$

Put $\sin x=t$, $\cos x\,dx=dt$:
$$y = e^{-2\sin x}\left(\frac{2+3\sin x}{-2}-\frac34\right)+c.$$

Using $x=0$: $c=0$. At $x=\pi/6$: $y=-\dfrac52$.

Solutions · 1