75.7.22 problem 197

Internal problem ID [16736]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 197
Date solved : Thursday, March 13, 2025 at 08:39:44 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.005 (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: 0.3 (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: 0.414 (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)
 

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