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 : Monday, January 27, 2025 at 05:09:15 AM
CAS classification : [_separable]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[x^2*y[x]*D[y[x],x]== (y[x]^2-1)^(3/2),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*}