29.23.12 problem 643

Internal problem ID [5234]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 643
Date solved : Sunday, March 30, 2025 at 07:06:12 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x \left (2 x^{2}+y^{2}\right ) y^{\prime }&=\left (2 x^{2}+3 y^{2}\right ) y \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{2 c_1} \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{-4 c_1}}{x^{4} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{-4 c_1}}{x^{4}}\right )}}\, x^{3} \]

✓ Mathematica. Time used: 7.229 (sec). Leaf size: 65

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

\begin{align*} y(x)\to -\frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-3-2 c_1}}{x^4}\right )}} \\ y(x)\to \frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-3-2 c_1}}{x^4}\right )}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.849 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = x^{3} e^{- 2 C_{1} + \frac {W\left (\frac {2 e^{4 C_{1}}}{x^{4}}\right )}{2}} \]