Internal problem ID [8442]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 864.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, `2nd type`, `class C`], [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {y^{\prime }-\frac {y \left ({\mathrm e}^{-\frac {x^{2}}{2}} x y+x \,{\mathrm e}^{-\frac {x^{2}}{4}}+2 y^{2} {\mathrm e}^{-\frac {3 x^{2}}{4}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2 \,{\mathrm e}^{-\frac {x^{2}}{4}} y+2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 186
dsolve(diff(y(x),x) = y(x)*(exp(-1/4*x^2)^2*x*y(x)+exp(-1/4*x^2)*x+2*y(x)^2*exp(-3/4*x^2))*exp(1/4*x^2)/(2*y(x)*exp(-1/4*x^2)+2),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {\left ({\mathrm e}^{-\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}\, {\mathrm e}^{\frac {x^{2}}{2}}+{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{2}}-{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{2}}-{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}} \\ y \left (x \right ) = \frac {\left ({\mathrm e}^{-\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}\, {\mathrm e}^{\frac {x^{2}}{2}}-{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{2}}-{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {-2 x +c_{1}}+{\mathrm e}^{-\frac {x^{2}}{4}} {\mathrm e}^{\frac {x^{2}}{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.094 (sec). Leaf size: 103
DSolve[y'[x] == (E^(x^2/4)*y[x]*(x/E^(x^2/4) + (x*y[x])/E^(x^2/2) + (2*y[x]^2)/E^((3*x^2)/4)))/(2 + (2*y[x])/E^(x^2/4)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\frac {x^2}{2}}}{-e^{\frac {x^2}{4}}+\sqrt {e^{\frac {x^2}{2}} (-2 x+1+c_1)}} \\ y(x)\to -\frac {e^{\frac {x^2}{2}}}{e^{\frac {x^2}{4}}+\sqrt {e^{\frac {x^2}{2}} (-2 x+1+c_1)}} \\ y(x)\to 0 \\ \end{align*}