85.10.4 problem 3

Internal problem ID [22491]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 37
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:41:20 PM
CAS classification : [_separable]

\begin{align*} x^{3} {\mathrm e}^{2 x^{2}+3 y^{2}}-y^{3} {\mathrm e}^{-x^{2}-2 y^{2}} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 79

ode:=x^3*exp(2*x^2+3*y(x)^2)-y(x)^3*exp(-x^2-2*y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-5-5 \operatorname {LambertW}\left (\frac {25 \left (3 x^{2} {\mathrm e}^{3 x^{2}}+18 c_1 -{\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-1}}{9}\right )}}{5} \\ y &= \frac {\sqrt {-5-5 \operatorname {LambertW}\left (\frac {25 \left (3 x^{2} {\mathrm e}^{3 x^{2}}+18 c_1 -{\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-1}}{9}\right )}}{5} \\ \end{align*}

✓ Mathematica. Time used: 60.174 (sec). Leaf size: 94

ode=x^3*Exp[2*x^2+3*y[x]^2] - y[x]^3*Exp[ -x^2-2*y[x]^2]*D[y[x],x]== 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {-1-W\left (\frac {25 \left (e^{3 x^2} \left (3 x^2-1\right )+18 c_1\right )}{9 e}\right )}}{\sqrt {5}}\\ y(x)&\to \frac {\sqrt {-1-W\left (\frac {25 \left (e^{3 x^2} \left (3 x^2-1\right )+18 c_1\right )}{9 e}\right )}}{\sqrt {5}} \end{align*}

✓ Sympy. Time used: 4.513 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*exp(2*x**2 + 3*y(x)**2) - y(x)**3*exp(-x**2 - 2*y(x)**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {5} \sqrt {- W\left (C_{1} + \frac {25 x^{2} e^{3 x^{2} - 1}}{3} - \frac {25 e^{3 x^{2} - 1}}{9}\right ) - 1}}{5}, \ y{\left (x \right )} = \frac {\sqrt {5} \sqrt {- W\left (C_{1} + \frac {25 x^{2} e^{3 x^{2} - 1}}{3} - \frac {25 e^{3 x^{2} - 1}}{9}\right ) - 1}}{5}\right ] \]