81.6.24 problem 7-23

Internal problem ID [21570]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 7. Linear Differential Equations. Page 101.
Problem number : 7-23
Date solved : Thursday, October 02, 2025 at 07:49:52 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime }&=\frac {1}{x^{2} y^{3}+y x} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 78

ode:=diff(y(x),x) = 1/(x*y(x)+x^2*y(x)^3); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.247 (sec). Leaf size: 76

ode=D[y[x],x]==1/(x*y[x]+x^2*y[x]^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.890 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - 1/(x**2*y(x)**3 + x*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \left (y^{2}{\left (x \right )} - 2\right ) \sqrt {e^{y^{2}{\left (x \right )}}} - \frac {\sqrt {e^{y^{2}{\left (x \right )}}}}{x} = 0 \]