89.30.16 problem 18

Internal problem ID [24933]
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 : 18
Date solved : Thursday, October 02, 2025 at 11:36:13 PM
CAS classification : [_Clairaut]

\begin{align*} y&=x y^{\prime }+{y^{\prime }}^{n} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 34

ode:=y(x) = x*diff(y(x),x)+diff(y(x),x)^n; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (-n +1\right ) \left (\left (-\frac {x}{n}\right )^{\frac {1}{n -1}}\right )^{n} \\ y &= c_1 x +c_1^{n} \\ \end{align*}

✓ Mathematica. Time used: 1.498 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to c_1{}^n+c_1 x\\ y(x)&\to 0^n \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [_X0, _X0**n] 
No algorithms are implemented to solve equat