29.19.3 problem 516

Internal problem ID [5112]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 516
Date solved : Monday, January 27, 2025 at 10:12:00 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y y^{\prime }+x^{2} {\mathrm e}^{-\frac {2 y}{x}}-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 18

dsolve(x*y(x)*diff(y(x),x)+x^2*exp(-2*y(x)/x)-y(x)^2 = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (\operatorname {LambertW}\left (-4 \left (\ln \left (x \right )+c_{1} \right ) {\mathrm e}^{-1}\right )+1\right ) x}{2} \]

✓ Solution by Mathematica

Time used: 60.184 (sec). Leaf size: 25

DSolve[x y[x] D[y[x],x]+x^2 Exp[(-2 y[x])/x]-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} x \left (1+W\left (\frac {4 (-\log (x)+c_1)}{e}\right )\right ) \]