47.1.6 problem 6

Internal problem ID [7387]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 6
Date solved : Sunday, March 30, 2025 at 11:56:16 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

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

\[ \ln \left (x \right )-\sqrt {1+y^{2}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.24 (sec). Leaf size: 65

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

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

✓ Sympy. Time used: 1.050 (sec). Leaf size: 46

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

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