29.23.18 problem 649

Internal problem ID [5240]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 649
Date solved : Tuesday, March 04, 2025 at 08:52:02 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.811 (sec). Leaf size: 1755

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.288 (sec). Leaf size: 873

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

\begin{align*} y(x)\to -\sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to \sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to -\frac {1}{2} \sqrt {-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}-3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to \frac {1}{2} \sqrt {-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}-3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to -\frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}+3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ y(x)\to \frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}+3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\ \end{align*}

✗ Sympy

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

Timed Out