23.1.658 problem 653

Internal problem ID [5265]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 653
Date solved : Tuesday, September 30, 2025 at 12:05:11 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.138 (sec). Leaf size: 89

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

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= \frac {\sqrt {-2-2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= -\frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ y &= \frac {\sqrt {-2+2 \sqrt {4 c_1 \,x^{2}+1}}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 39.566 (sec). Leaf size: 277

ode=x(x^2+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 {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-x^2+\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\\ y(x)&\to \sqrt {-\frac {x^2}{2}+\frac {1}{2} \sqrt {x^4+4 e^{2 c_1} x^6}}\\ y(x)&\to -\frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt {x^4}-x^2}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 4.767 (sec). Leaf size: 116

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 + 3*y(x)**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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