23.1.693 problem 688

Internal problem ID [5300]
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 : 688
Date solved : Tuesday, September 30, 2025 at 12:19:20 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (3 x^{2}+2 y^{2}\right ) y y^{\prime }+x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.145 (sec). Leaf size: 137

ode:=(3*x^2+2*y(x)^2)*y(x)*diff(y(x),x)+x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-8 x^{2} c_1^{2}-2 \sqrt {8 x^{2} c_1^{2}+1}+2}}{4 c_1} \\ y &= \frac {\sqrt {-8 x^{2} c_1^{2}-2 \sqrt {8 x^{2} c_1^{2}+1}+2}}{4 c_1} \\ y &= -\frac {\sqrt {-8 x^{2} c_1^{2}+2 \sqrt {8 x^{2} c_1^{2}+1}+2}}{4 c_1} \\ y &= \frac {\sqrt {-8 x^{2} c_1^{2}+2 \sqrt {8 x^{2} c_1^{2}+1}+2}}{4 c_1} \\ \end{align*}

✓ Mathematica. Time used: 17.399 (sec). Leaf size: 253

ode=(3*x^2+2*y[x]^2)*y[x]*D[y[x],x]+x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\\ y(x)&\to \frac {\sqrt {-4 x^2-\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\\ y(x)&\to \frac {\sqrt {-4 x^2+\sqrt {8 e^{2 c_1} x^2+e^{4 c_1}}+e^{2 c_1}}}{2 \sqrt {2}}\\ y(x)&\to \text {Undefined}\\ y(x)&\to -\frac {\sqrt {-x^2}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^2}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 7.098 (sec). Leaf size: 119

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

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