How do I solve $\frac{dy}{dx}=5xy + \sin x$ explicitly? With $y(0) = 1$. I am asked to use an integrating factor. What I did:
$\frac{dy}{dx}-5xy = \sin x \\ \text{Integrating factor:} \ e^{\int{-5x\ dx}} = e^{-\frac{5}{2}x^2} \\ \frac{d}{dx}\left[e^{-\frac{5}{2}x^2}y\right] = e^{-\frac{5}{2}x^2}\sin x \\ e^{-\frac{5}{2}x^2}y = \int e^{-\frac{5}{2}x^2}\sin x \ dx$
How would I proceed from there? Edit: $y(0) = 1$.
Also, when does the scalar ODE (above) have a unique solution?