29.23.9 problem 640

Internal problem ID [5231]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 640
Date solved : Tuesday, March 04, 2025 at 08:40:43 PM
CAS classification : [[_homogeneous, `class D`], _rational]

\begin{align*} x \left (x^{2}+y^{2}\right ) y^{\prime }&=\left (x^{2}+x^{4}+y^{2}\right ) y \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 36

ode:=x*(x^2+y(x)^2)*diff(y(x),x) = (x^2+x^4+y(x)^2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {x^{2}}{2}+c_{1}} x}{\sqrt {\frac {{\mathrm e}^{x^{2}+2 c_{1}}}{\operatorname {LambertW}\left ({\mathrm e}^{x^{2}+2 c_{1}}\right )}}} \]

✓ Mathematica. Time used: 4.507 (sec). Leaf size: 49

ode=x(x^2+y[x]^2)D[y[x],x]==(x^2+x^4+y[x]^2)y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + y(x)**2)*Derivative(y(x), x) - (x**4 + x**2 + y(x)**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**4 + x**2 + y(x)**2)*y(x)/(x*(x**2 + y(x)**2)) cannot be solved by the factorable group method