63.1.69 problem 97

Internal problem ID [15509]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 97
Date solved : Thursday, October 02, 2025 at 10:19:12 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Clairaut]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 27

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

\begin{align*} y &= -2 \sqrt {x} \\ y &= 2 \sqrt {x} \\ y &= c_1 x +\frac {1}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to c_1 x+\frac {1}{c_1}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to -2 \sqrt {x}\\ y(x)&\to 2 \sqrt {x} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*x + y(x)**2) + y(x))/(2*x) cannot