20.24.11 problem Problem 11

Internal problem ID [3996]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 11
Date solved : Sunday, March 30, 2025 at 02:14:04 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

Order:=6; 
ode:=(-4*x^2+1)*diff(diff(y(x),x),x)-20*x*diff(y(x),x)-16*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+8 x^{2}+\frac {128}{3} x^{4}\right ) y \left (0\right )+\left (30 x^{5}+6 x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

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

ode=(1-4*x^2)*D[y[x],{x,2}]-20*x*D[y[x],x]-16*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (30 x^5+6 x^3+x\right )+c_1 \left (\frac {128 x^4}{3}+8 x^2+1\right ) \]

✓ Sympy. Time used: 0.872 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-20*x*Derivative(y(x), x) + (1 - 4*x**2)*Derivative(y(x), (x, 2)) - 16*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {128 x^{4}}{3} + 8 x^{2} + 1\right ) + C_{1} x \left (6 x^{2} + 1\right ) + O\left (x^{6}\right ) \]