72.8.29 problem 29

Internal problem ID [19513]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 29
Date solved : Thursday, October 02, 2025 at 04:34:22 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 34

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]
 

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

✓ Sympy. Time used: 1.194 (sec). Leaf size: 19

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} + \log {\left (e^{\frac {x^{2}}{y^{2}{\left (x \right )}}} + 1 \right )} \]