23.2.248 problem 254

Internal problem ID [5603]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 254
Date solved : Sunday, October 12, 2025 at 01:25:28 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.241 (sec). Leaf size: 247

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

\begin{align*} y &= \left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \\ y &= \left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \\ y &= -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ y &= -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ y &= 0 \\ \int _{\textit {\_b}}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} +\frac {\ln \left (x \right )}{2}-c_1 &= 0 \\ \int _{\textit {\_b}}^{y}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} -\frac {\ln \left (x \right )}{2}-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.488 (sec). Leaf size: 194

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

\begin{align*} y(x)&\to \frac {\sqrt [3]{a+(-1+c_1) x^3}}{\sqrt [3]{1-\frac {1}{c_1}}}\\ y(x)&\to 0\\ y(x)&\to \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}-x^{3/2}\right )^2}\\ y(x)&\to \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\left (\sqrt {a}+x^{3/2}\right )^2} \end{align*}

✗ Sympy

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

Timed Out