89.30.15 problem 17

Internal problem ID [24932]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 243
Problem number : 17
Date solved : Thursday, October 02, 2025 at 11:36:13 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+1&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 66

ode:=x*diff(y(x),x)^3-y(x)*diff(y(x),x)^2+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 69

ode=x*D[y[x],x]^3-y[x]*D[y[x],x]^2+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)**3 - y(x)*Derivative(y(x), x)**2 + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out