30.7.2 problem 5 (e)

Internal problem ID [7577]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problem. (F) Clairaut equation. page 85
Problem number : 5 (e)
Date solved : Tuesday, September 30, 2025 at 04:54:26 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 68

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 71

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

\begin{align*} y(x)&\to c_1 x+\frac {2}{c_1{}^2}\\ y(x)&\to -3 \sqrt [3]{-\frac {1}{2}} x^{2/3}\\ y(x)&\to \frac {3 x^{2/3}}{\sqrt [3]{2}}\\ y(x)&\to \frac {3 (-1)^{2/3} x^{2/3}}{\sqrt [3]{2}} \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 + 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out