31.1.19 problem 10.1

Internal problem ID [5717]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 10.1
Date solved : Monday, January 27, 2025 at 01:11:21 PM
CAS classification : [_separable]

\begin{align*} \left (-x^{2}+1\right ) z^{\prime }-x z&=a x z^{2} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 22

dsolve((1-x^2)*diff(z(x),x)-x*z(x)=a*x*z(x)^2,z(x), singsol=all)
 

\[ z \left (x \right ) = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_{1} -a} \]

✓ Solution by Mathematica

Time used: 3.841 (sec). Leaf size: 47

DSolve[(1-x^2)*D[z[x],x]-x*z[x]==a*x*z[x]^2,z[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} z(x)\to -\frac {e^{c_1}}{-\sqrt {1-x^2}+a e^{c_1}} \\ z(x)\to 0 \\ z(x)\to -\frac {1}{a} \\ \end{align*}