47.2.55 problem 51

Internal problem ID [7471]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 51
Date solved : Sunday, March 30, 2025 at 12:09:46 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 33

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

\[ y = \frac {{\mathrm e}^{-2 c_1} x}{\sqrt {-\frac {x^{4} {\mathrm e}^{-4 c_1}}{\operatorname {LambertW}\left (-x^{4} {\mathrm e}^{-4 c_1}\right )}}} \]

✓ Mathematica. Time used: 4.294 (sec). Leaf size: 60

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

\begin{align*} y(x)\to -\frac {i \sqrt {W\left (-e^{-2 c_1} x^4\right )}}{x} \\ y(x)\to \frac {i \sqrt {W\left (-e^{-2 c_1} x^4\right )}}{x} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.084 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = x e^{2 C_{1} - \frac {W\left (- x^{4} e^{4 C_{1}}\right )}{2}} \]