The solution of the differential equation $x \frac{d y}{d x}=2 y+x^3 e^x$, where $y=0$ when $x=1$ is

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Professor PC Thomas Classes · Answer

Linear DE: $\frac{dy}{dx} - \frac{2}{x}y = x^2 e^x$. IF $= 1/x^2$. Solution: $y/x^2 = e^x + c$. Given $y(1)=0 \implies c = -e \implies y = x^2(e^x - e)$.