Internal problem ID [3729]
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`]]
\[ \boxed {\left (x \,{\mathrm e}^{-x}-2 y\right ) y^{\prime }+\left ({\mathrm e}^{-x}+x \,{\mathrm e}^{-x}-2 y\right ) y=2 x \,{\mathrm e}^{-2 x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 88
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 \left (x \right ) = -\frac {\left (-{\mathrm e}^{x} x +\sqrt {-3 x^{2} {\mathrm e}^{2 x}+4 c_{1} {\mathrm e}^{2 x}}\right ) {\mathrm e}^{-2 x}}{2} y \left (x \right ) = \frac {{\mathrm e}^{-2 x} \left ({\mathrm e}^{x} x +\sqrt {-3 x^{2} {\mathrm e}^{2 x}+4 c_{1} {\mathrm e}^{2 x}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 33.003 (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*}