6.12.4 problem 4

Internal problem ID [1858]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:20:37 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-8*x*diff(y(x),x)-12*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (15 x^{4}+6 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

ode=(1-x^2)*D[y[x],{x,2}]-8*x*D[y[x],x]-12*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (7 x^5+\frac {10 x^3}{3}+x\right )+c_1 \left (15 x^4+6 x^2+1\right ) \]

✓ Sympy. Time used: 0.276 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*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_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (15 x^{4} + 6 x^{2} + 1\right ) + C_{1} x \left (\frac {10 x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]