87.14.19 problem 19

Internal problem ID [23538]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:42:47 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {5}{2} x^{3}-\frac {3}{2} x \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 53

ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+12*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (-\frac {5}{3} x^{3}+x \right )+c_2 \left (-\frac {1}{9}+\frac {\left (5 x^{3}-3 x \right ) \ln \left (x -1\right )}{24}+\frac {\left (-5 x^{3}+3 x \right ) \ln \left (x +1\right )}{24}+\frac {5 x^{2}}{12}\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 59

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+12*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} c_1 x \left (5 x^2-3\right )+c_2 \left (-\frac {5 x^2}{2}-\frac {1}{4} \left (5 x^2-3\right ) x (\log (1-x)-\log (x+1))+\frac {2}{3}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False