44.5.14 problem 14

Internal problem ID [7076]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 14
Date solved : Sunday, March 30, 2025 at 11:38:38 AM
CAS classification : [_separable]

\begin{align*} x \sqrt {1+y^{2}}&=y \sqrt {x^{2}+1}\, y^{\prime } \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: 0.277 (sec). Leaf size: 75

ode=x*Sqrt[1+y[x]^2]==y[x]*Sqrt[1+x^2]*D[y[x],x]; 
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: 1.221 (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)
 

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