23.1.597 problem 590

Internal problem ID [5204]
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 : 590
Date solved : Tuesday, September 30, 2025 at 11:54:55 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 8 x^{3} y y^{\prime }+3 x^{4}-6 x^{2} y^{2}-y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 54

ode:=8*x^3*y(x)*diff(y(x),x)+3*x^4-6*x^2*y(x)^2-y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x \sqrt {-\left (c_1 x -1\right ) \left (c_1 x +3\right )}}{c_1 x -1} \\ y &= -\frac {x \sqrt {-\left (c_1 x -1\right ) \left (c_1 x +3\right )}}{c_1 x -1} \\ \end{align*}

✓ Mathematica. Time used: 4.363 (sec). Leaf size: 160

ode=8*x^3*y[x]*D[y[x],x]+3*x^4 -6*x^2*y[x]^2 -y[x]^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {-x^2 \left (3+e^{8 c_1} x\right )}}{\sqrt {-1+e^{8 c_1} x}}\\ y(x)&\to \frac {\sqrt {-x^2 \left (3+e^{8 c_1} x\right )}}{\sqrt {-1+e^{8 c_1} x}}\\ y(x)&\to -i \sqrt {3} \sqrt {-x^2}\\ y(x)&\to i \sqrt {3} \sqrt {-x^2}\\ y(x)&\to \frac {x^{5/2}}{\sqrt {-x^3}}\\ y(x)&\to \frac {\sqrt {-x^3}}{\sqrt {x}} \end{align*}

✓ Sympy. Time used: 1.047 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**4 + 8*x**3*y(x)*Derivative(y(x), x) - 6*x**2*y(x)**2 - y(x)**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \sqrt {\frac {3 C_{1} - x}{C_{1} + x}}, \ y{\left (x \right )} = - x \sqrt {\frac {3 C_{1} - x}{C_{1} + x}}\right ] \]