12.3.5 problem 6

Internal problem ID [1582]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 12:40:42 PM
CAS classification : [_separable]

\begin{align*} x^{2} y y^{\prime }&=\left (y^{2}-1\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=x^2*y(x)*diff(y(x),x) = (-1+y(x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {1}{x}+\frac {\left (y-1\right ) \left (y+1\right )}{\left (y^{2}-1\right )^{{3}/{2}}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.596 (sec). Leaf size: 111

ode=x^2*y[x]*D[y[x],x]== (y[x]^2-1)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 15.829 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x)*Derivative(y(x), x) - (y(x)**2 - 1)**(3/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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