83.4.4 problem 1 (d)

Internal problem ID [21931]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. V at page 42
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 08:14:59 PM
CAS classification : [_exact]

\begin{align*} x^{2}+y \,{\mathrm e}^{2 y}+\left (2 y x +x \right ) {\mathrm e}^{2 y} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 19

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

\[ y = \frac {\operatorname {LambertW}\left (-\frac {2 \left (x^{3}+3 c_1 \right )}{3 x}\right )}{2} \]

✓ Mathematica. Time used: 60.122 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 1.091 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = \frac {W\left (\frac {2 \left (C_{1} - x^{3}\right )}{3 x}\right )}{2} \]