Let y=y(x) solve dydx=2(y+2 x-5)x-2 x, y(0)=7.

Question

Let $y=y(x)$ solve $\dfrac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$, $y(0)=7$. Then $y(\pi)$ is:

Accepted Answer

Let $t=y+2\sin x-5$: $\dfrac{dt}{dx}=2tx \implies \ln t = x^2+C$.

Using $y(0)=7$: $\ln2=C$. At $x=\pi$: $\ln(y(\pi)-5)=\pi^2+\ln2 \implies y(\pi)=2e^{\pi^2}+5$.

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