15.7.21 problem 21

Internal problem ID [3002]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 21
Date solved : Tuesday, March 04, 2025 at 03:39:50 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} 1+x y \left (1+x y^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.261 (sec). Leaf size: 66

ode:=1+x*y(x)*(1+x*y(x)^2)*diff(y(x),x) = 0; 
ic:=y(1) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✗ Mathematica

ode=1+x*y[x]*(1+x*y[x]^2)*D[y[x],x]==0; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x*y(x)**2 + 1)*y(x)*Derivative(y(x), x) + 1,0) 
ics = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)