Internal problem ID [8402]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 822.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (4 \,{\mathrm e}^{-x^{2}}-4 \,{\mathrm e}^{-x^{2}} x^{2}+4 y^{2}-4 x^{2} {\mathrm e}^{-x^{2}} y+{\mathrm e}^{-2 x^{2}} x^{4}\right ) x}{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve(diff(y(x),x) = 1/4*(4*exp(-x^2)-4*x^2*exp(-x^2)+4*y(x)^2-4*x^2*exp(-x^2)*y(x)+x^4*exp(-x^2)^2)*x,y(x), singsol=all)
\[ y \relax (x ) = \frac {x^{2} {\mathrm e}^{-x^{2}}}{2}+\frac {1}{c_{1}-\frac {x^{2}}{2}} \]
✓ Solution by Mathematica
Time used: 0.502 (sec). Leaf size: 50
DSolve[y'[x] == (x*(4/E^x^2 - (4*x^2)/E^x^2 + x^4/E^(2*x^2) - (4*x^2*y[x])/E^x^2 + 4*y[x]^2))/4,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-x^2} x^2+\frac {1}{-\frac {x^2}{2}+c_1} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2 \\ \end{align*}