63.1.70 problem 98

Internal problem ID [15510]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 98
Date solved : Thursday, October 02, 2025 at 10:19:13 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 72

ode:=y(x) = x*diff(y(x),x)-1/diff(y(x),x)^2; 
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: 0.008 (sec). Leaf size: 71

ode=y[x]==x*D[y[x],x]-1/(D[y[x],x])^2; 
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) + y(x) + Derivative(y(x), x)**(-2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out