2.4   ODE No. 4

1. Problem in Latex
2. Mathematica input
3. Maple input

$$-e^{-x^2}x+y^{\prime }(x)+2xy(x)=0$$ ✓ Mathematica : cpu = 0.0133891 (sec), leaf count = 30

$$\left\{\{y(x)\to \frac{1}{2}{e}^{-x^2}{x}^2+c_1{e}^{-x^2}\}\right\}$$ ✓ Maple : cpu = 0.005 (sec), leaf count = 18

$$\{y\left(x\right)=(\frac{x^2}{2}+\mathit{\_}\mathit{C}\mathit{1}){\mathrm{e}}^{-x^2}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& \frac{dy}{dx}+2xy\left(x\right)=e^{-x^2}x\end{array}$$

Integrating factor $\mu =e^{\int 2xdx}=e^{x^2}$. Hence (1) becomes

$$\begin{array}{rl}\frac{d}{dx}\left(e^{x^2}y\left(x\right)\right)& =e^{x^2}{e}^{-x^2}x\\ \frac{d}{dx}\left(e^{x^2}y\left(x\right)\right)& =x\end{array}$$

Integrating both sides

$$\begin{array}{rl}{e}^{x^2}y\left(x\right)& =\frac{x^2}{2}+C\\ y\left(x\right)& =e^{-x^2}(\frac{x^2}{2}+C)\end{array}$$