89.32.20 problem 23

Internal problem ID [24980]
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. Miscellaneous Exercises at page 246
Problem number : 23
Date solved : Thursday, October 02, 2025 at 11:45:17 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} \left (y^{\prime }+1\right )^{2} \left (y-x y^{\prime }\right )&=1 \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 93

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 102

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

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

✗ Sympy

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

Timed Out