75.4.5 problem 50

Internal problem ID [16636]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 50
Date solved : Monday, March 31, 2025 at 03:02:40 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ \sqrt {x^{2}+1}+\sqrt {1+y^{2}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.279 (sec). Leaf size: 75

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

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

✓ Sympy. Time used: 1.149 (sec). Leaf size: 49

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

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