23.2.294 problem 313

Internal problem ID [5649]
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 : 313
Date solved : Tuesday, September 30, 2025 at 01:24:07 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 74

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

\begin{align*} y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y &= \frac {c_1^{3} x +a}{c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 86

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

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

✗ Sympy

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

Timed Out