Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}-2xy=2\left(2\sin x-5\right)x-2\cos x$

$IF=e^{-x^2}$

So, $y e^{-x^2}=\int e^{-x^2}\left(2x\left(2\sin x-5\right)-2\cos x\right) dx$

$\Rightarrow y·e^{-x^2}=e^{-x^2}\left(5-2\sin x\right)+C$

$y=5-2 \sin x+C·e^{x^2}$

Given at $x=0, y=7$

$\Rightarrow 7=5+C$

$\Rightarrow C=2$

So, $y=5-2 \sin x+2e^{x^2}$

Now at $x=\pi$

$y=5+2e^{{\pi }^2}$