Internal problem ID [3221]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 17
Problem number: 475.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {\left (x \,{\mathrm e}^{-x}-2 y\right ) y^{\prime }-2 \,{\mathrm e}^{-2 x} x +\left ({\mathrm e}^{-x}+x \,{\mathrm e}^{-x}-2 y\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 89
dsolve((x*exp(-x)-2*y(x))*diff(y(x),x) = 2*x*exp(-2*x)-(exp(-x)+x*exp(-x)-2*y(x))*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {{\mathrm e}^{-2 x} \left (x \,{\mathrm e}^{x}-\sqrt {-3 \,{\mathrm e}^{2 x} x^{2}+4 c_{1} {\mathrm e}^{2 x}}\right )}{2} \\ y \relax (x ) = \frac {{\mathrm e}^{-2 x} \left (x \,{\mathrm e}^{x}+\sqrt {-3 \,{\mathrm e}^{2 x} x^{2}+4 c_{1} {\mathrm e}^{2 x}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 32.457 (sec). Leaf size: 81
DSolve[(x Exp[-x]-2 y[x])y'[x]==2 x Exp[-2 x]-(Exp[-x]+x Exp[-x]-2 y[x])y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-2 x} \left (e^x x-\sqrt {e^{2 x} \left (-3 x^2+4 c_1\right )}\right ) \\ y(x)\to \frac {1}{2} e^{-2 x} \left (e^x x+\sqrt {e^{2 x} \left (-3 x^2+4 c_1\right )}\right ) \\ \end{align*}