Internal problem ID [8526]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 946.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left ({\mathrm e}^{-3 x^{2}} x^{6}-6 \,{\mathrm e}^{-2 x^{2}} x^{4} y+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-4 \,{\mathrm e}^{-2 x^{2}} x^{4}+8 x^{2} {\mathrm e}^{-x^{2}} y+8 \,{\mathrm e}^{-x^{2}} x^{2}+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 y^{3}-8 \,{\mathrm e}^{-x^{2}} y-8 \,{\mathrm e}^{-x^{2}}\right ) x}{-8 y+4 \,{\mathrm e}^{-x^{2}} x^{2}-8}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 100
dsolve(diff(y(x),x) = (-8*exp(-x^2)*y(x)+4*x^2*exp(-x^2)^2-8*exp(-x^2)+8*x^2*exp(-x^2)*y(x)-4*x^4*exp(-x^2)^2+8*x^2*exp(-x^2)-8*y(x)^3+12*x^2*exp(-x^2)*y(x)^2-6*y(x)*x^4*exp(-x^2)^2+x^6*exp(-x^2)^3)*x/(-8*y(x)+4*x^2*exp(-x^2)-8),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-x^{2}+c_{1}}\, {\mathrm e}^{-x^{2}} x^{2}-x^{2} {\mathrm e}^{-x^{2}}+2}{2 \sqrt {-x^{2}+c_{1}}-2} \\ y \relax (x ) = \frac {\sqrt {-x^{2}+c_{1}}\, {\mathrm e}^{-x^{2}} x^{2}+x^{2} {\mathrm e}^{-x^{2}}-2}{2 \sqrt {-x^{2}+c_{1}}+2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.019 (sec). Leaf size: 93
DSolve[y'[x] == (x*(-8/E^x^2 + (4*x^2)/E^(2*x^2) + (8*x^2)/E^x^2 - (4*x^4)/E^(2*x^2) + x^6/E^(3*x^2) - (8*y[x])/E^x^2 + (8*x^2*y[x])/E^x^2 - (6*x^4*y[x])/E^(2*x^2) + (12*x^2*y[x]^2)/E^x^2 - 8*y[x]^3))/(-8 + (4*x^2)/E^x^2 - 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-x^2} x^2+\frac {8}{-8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2-\frac {8}{8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2 \\ \end{align*}