81.3.1 problem 4-6

Internal problem ID [21500]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 4. Homogeneous differential equations.
Problem number : 4-6
Date solved : Thursday, October 02, 2025 at 07:42:13 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.131 (sec). Leaf size: 52

ode:=diff(y(x),x) = 2*x*y(x)*exp(x/y(x))^2/(y(x)^2+y(x)^2*exp(x/y(x))^2+2*x^2*exp(x/y(x))^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \operatorname {polylog}\left (2, -{\mathrm e}^{2 \,{\mathrm e}^{-\textit {\_Z}}}\right )+2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )+2 \,{\mathrm e}^{\textit {\_Z}} c_1 +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left (1+{\mathrm e}^{2 \,{\mathrm e}^{-\textit {\_Z}}}\right )\right )} x \]

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 68

ode=D[y[x],x] ==(2*x*y[x]*Exp[x/y[x]]^2 )/( y[x]^2+y[x]^2*Exp[x/y[x]]^2+2*x^2*Exp[x/y[x]]^2    ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {e^{\frac {2}{K[1]}} K[1]^2+K[1]^2+2 e^{\frac {2}{K[1]}}}{\left (1+e^{\frac {2}{K[1]}}\right ) K[1]^3}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.968 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*y(x)*exp(2*x/y(x))/(2*x**2*exp(2*x/y(x)) + y(x)**2*exp(2*x/y(x)) + y(x)**2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{2 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1} e^{2 u_{1}}}{e^{2 u_{1}} + 1}\, du_{1}} \]