29.17.16 problem 475

Internal problem ID [5073]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 17
Problem number : 475
Date solved : Monday, January 27, 2025 at 10:07:57 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} \left (x \,{\mathrm e}^{-x}-2 y\right ) y^{\prime }&=2 x \,{\mathrm e}^{-2 x}-\left ({\mathrm e}^{-x}+x \,{\mathrm e}^{-x}-2 y\right ) y \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 63

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 (x \,{\mathrm e}^{x}-\sqrt {{\mathrm e}^{2 x} \left (-3 x^{2}+4 c_{1} \right )}\right ) {\mathrm e}^{-2 x}}{2} \\ y \left (x \right ) &= \frac {\left (x \,{\mathrm e}^{x}+\sqrt {{\mathrm e}^{2 x} \left (-3 x^{2}+4 c_{1} \right )}\right ) {\mathrm e}^{-2 x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 33.019 (sec). Leaf size: 81

DSolve[(x Exp[-x]-2 y[x])D[y[x],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*}