54.2.16 problem 19

Internal problem ID [8548]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 19
Date solved : Sunday, March 30, 2025 at 01:17:28 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Maple. Time used: 0.043 (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: 0.953 (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