23.1.691 problem 686

Internal problem ID [5298]
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 : 686
Date solved : Tuesday, September 30, 2025 at 12:15:40 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 2 y^{3} y^{\prime }&=x^{3}-x y^{2} \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 769

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

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

✓ Mathematica. Time used: 60.08 (sec). Leaf size: 714

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

\begin{align*} y(x)&\to -\frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}}\\ y(x)&\to -\frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\\ y(x)&\to \frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\\ y(x)&\to -\frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\\ y(x)&\to \frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \end{align*}

✗ Sympy

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

Timed Out