82.12.15 problem Ex. 15

Internal problem ID [18701]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 15
Date solved : Thursday, March 13, 2025 at 12:39:27 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 \left (x \right ) &= \sqrt {c_{1} x +x^{2}-1} \\ y \left (x \right ) &= -\sqrt {c_{1} x +x^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 0.306 (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: 0.404 (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)
 

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