75.2.8 problem 8

Internal problem ID [19902]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 8
Date solved : Thursday, October 02, 2025 at 05:00:43 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 +\ln \left (x +\sqrt {x^{2}-1}\right ) c_2 \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 23

ode=(1-x^2)*D[y[x],{x,2}]-x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \log \left (\sqrt {x^2-1}+x\right )+c_2 \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 17

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

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