67.24.8 problem 34.5 (h)

Internal problem ID [16975]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.5 (h)
Date solved : Thursday, October 02, 2025 at 01:41:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-4\right ) y^{\prime \prime }+\left (x^{2}+x -6\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {5 \left (x -2\right )^{2}}{8}+\frac {\left (x -2\right )^{3}}{96}+\frac {49 \left (x -2\right )^{4}}{768}-\frac {37 \left (x -2\right )^{5}}{15360}\right ) y \left (2\right )+\left (x -2-\frac {5 \left (x -2\right )^{3}}{24}+\frac {\left (x -2\right )^{4}}{192}+\frac {47 \left (x -2\right )^{5}}{3840}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 78

ode=(x^2-4)*D[y[x],{x,2}]+(x^2+x-6)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (-\frac {37 (x-2)^5}{15360}+\frac {49}{768} (x-2)^4+\frac {1}{96} (x-2)^3-\frac {5}{8} (x-2)^2+1\right )+c_2 \left (\frac {47 (x-2)^5}{3840}+\frac {1}{192} (x-2)^4-\frac {5}{24} (x-2)^3+x-2\right ) \]

✓ Sympy. Time used: 0.327 (sec). Leaf size: 70

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

\[ y{\left (x \right )} = - \frac {\left (x - 2\right )^{3} r{\left (2 \right )}}{12} - \frac {3 \left (x - 2\right )^{4} r{\left (2 \right )}}{32} + \frac {13 \left (x - 2\right )^{5} r{\left (2 \right )}}{1920} + C_{2} \left (x + \frac {47 \left (x - 2\right )^{5}}{3840} + \frac {\left (x - 2\right )^{4}}{192} - \frac {5 \left (x - 2\right )^{3}}{24} - 2\right ) + C_{1} + O\left (x^{6}\right ) \]