Question

Solve: dy/dx + 2xy= e^-x^2

Answer 1

$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x} + 2xy = e^{-x^2} \\ \textsf{For the ODE}\hspace{0.1cm} \frac{\mathrm{d}y}{\mathrm{d}x} + Py = Q, \\ \textsf{Where P and Q are functions} \\\textsf{of x, it has the solution} \\ y IF = \int Q\cdot IF\hspace{0.1cm} \mathrm{d}x, \\ \textsf{Where IF is the integrating}\\\textsf{factor}\\ IF = e^{\int P \hspace{0.1cm} \mathrm{d}x} = e^{\int 2x \hspace{0.1cm} \mathrm{d}x} = e^{x^2} \\ \therefore ye^{x^2} = \int e^{-x^2} \cdot e^{x^2} \hspace{0.1cm} \mathrm{d}x\\ = \int e^0 \mathrm{d}x = \int \mathrm{d}x = x + C \\ \therefore ye^{x^2} = x + C, \Rightarrow\\ y = e^{-x^2}(x + C) \\ \textit{Where C is an arbitrary constant}\\ \therefore y = e^{-x^2}(x + C)\\ \textsf{is the solution to the differential}\\\textsf{equation}$