Let y(x) solve (2+ x)(1+y)dydx= x.

Question

Let $y(x)$ solve $(2+\sin x)(1+y)\dfrac{dy}{dx}=\cos x$. If $y(0)=2$, then $y\left(\dfrac{\pi}{2}\right)$ equals:

Accepted Answer

$$\int\frac{dy}{1+y}=\int\frac{\cos x}{2+\sin x}\,dx \implies 1+y=C(2+\sin x).$$

Using $(0,2)$: $C=3/2$. At $x=\pi/2$: $1+y=\dfrac32\cdot3=\dfrac92 \implies y=\dfrac72$.

Solutions · 1