85.9.8 problem 1 (h)

Internal problem ID [22482]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 37
Problem number : 1 (h)
Date solved : Thursday, October 02, 2025 at 08:40:50 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.183 (sec). Leaf size: 75

ode=x*Sqrt[1+y[x]^2]==y[x]*Sqrt[1+x^2]*D[y[x],{x,1}]; 
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: 0.740 (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)
 

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