77.1.4 problem 15 (page 27)

Internal problem ID [17823]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 15 (page 27)
Date solved : Monday, March 31, 2025 at 04:33:24 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.352 (sec). Leaf size: 37

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

\[ y = \sqrt {3-2 \sqrt {x^{2}+1}\, \sqrt {2}+x^{2}+2 \sqrt {2}-2 \sqrt {x^{2}+1}} \]

✓ Mathematica. Time used: 0.305 (sec). Leaf size: 95

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

\begin{align*} y(x)\to \sqrt {x^2-2 \sqrt {2} \sqrt {x^2+1}-2 \sqrt {x^2+1}+2 \sqrt {2}+3} \\ y(x)\to \sqrt {x^2+2 \sqrt {2} \sqrt {x^2+1}-2 \sqrt {x^2+1}-2 \sqrt {2}+3} \\ \end{align*}

✗ Sympy

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 = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants