25.1.19 problem 10.1

Internal problem ID [6852]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 10.1
Date solved : Tuesday, September 30, 2025 at 03:55:50 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

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

\[ z = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_1 -a} \]

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 57

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

\begin{align*} z(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (a K[1]+1)}dK[1]\&\right ]\left [-\frac {1}{2} \log \left (1-x^2\right )+c_1\right ]\\ z(x)&\to 0\\ z(x)&\to -\frac {1}{a} \end{align*}

✓ Sympy. Time used: 1.000 (sec). Leaf size: 44

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

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