87.6.4 problem 4

Internal problem ID [23321]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 53
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:32:02 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.189 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.272 (sec). Leaf size: 29

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

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} x + x^{2} - 1}, \ y{\left (x \right )} = \sqrt {C_{1} x + x^{2} - 1}\right ] \]