89.32.17 problem 20

Internal problem ID [24977]
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 : 20
Date solved : Thursday, October 02, 2025 at 11:45:16 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

\begin{align*} {y^{\prime }}^{2} x^{2}-\left (1+2 y x \right ) y^{\prime }+1+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 42

ode:=x^2*diff(y(x),x)^2-(2*x*y(x)+1)*diff(y(x),x)+y(x)^2+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {4 x^{2}-1}{4 x} \\ y &= c_1 x -\sqrt {c_1 -1} \\ y &= c_1 x +\sqrt {c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 0.914 (sec). Leaf size: 66

ode=x^2*D[y[x],x]^2-(2*x*y[x]+1)*D[y[x],x]+y[x]^2+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x+e^{-2 c_1} x+e^{-c_1}\\ y(x)&\to x+\frac {1}{4} e^{-2 c_1} x+\frac {e^{-c_1}}{2}\\ y(x)&\to x\\ y(x)&\to x-\frac {1}{4 x} \end{align*}

✗ Sympy

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

Timed Out