87.24.16 problem 16

Internal problem ID [23798]
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 5. Series solutions of second order linear equations. Exercise at page 218
Problem number : 16
Date solved : Thursday, October 02, 2025 at 09:45:18 PM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

\begin{align*} -1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 42

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+12*y(x) = 0; 
dsolve(ode,y(x),type='series',x=-1);
 

\[ y = \left (\frac {25}{2} \left (x +1\right )-\frac {203}{8} \left (x +1\right )^{2}+\frac {293}{24} \left (x +1\right )^{3}-\frac {35}{64} \left (x +1\right )^{4}-\frac {7}{80} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right ) c_2 +\left (1-6 \left (x +1\right )+\frac {15}{2} \left (x +1\right )^{2}-\frac {5}{2} \left (x +1\right )^{3}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right ) \left (c_2 \ln \left (x +1\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 109

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+12*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to c_1 \left (-\frac {5}{2} (x+1)^3+\frac {15}{2} (x+1)^2-6 (x+1)+1\right )+c_2 \left (-\frac {7}{80} (x+1)^5-\frac {35}{64} (x+1)^4+\frac {293}{24} (x+1)^3-\frac {203}{8} (x+1)^2+\frac {25 (x+1)}{2}+\left (-\frac {5}{2} (x+1)^3+\frac {15}{2} (x+1)^2-6 (x+1)+1\right ) \log (x+1)\right ) \]

✓ Sympy. Time used: 2.599 (sec). Leaf size: 604

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,hint="2nd_power_series_regular",x0=-1,n=6)
 

\[ \text {Solution too large to show} \]