77.1.22 problem 38 (page 41)

Internal problem ID [17833]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 38 (page 41)
Date solved : Thursday, March 13, 2025 at 10:58:46 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 3.628 (sec). Leaf size: 53

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

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

✓ Sympy. Time used: 0.581 (sec). Leaf size: 42

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

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