84.29.1 problem 18.5 (a)

Internal problem ID [22286]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 18. Linear differential equations with variable coefficients. Solved problems. Page 94
Problem number : 18.5 (a)
Date solved : Thursday, October 02, 2025 at 08:37:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

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

\[ y = \left (1+\frac {\left (x -1\right )^{2}}{6}+\frac {\left (x -1\right )^{3}}{27}+\frac {17 \left (x -1\right )^{4}}{648}+\frac {2 \left (x -1\right )^{5}}{135}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{3}}{18}+\frac {\left (x -1\right )^{4}}{54}+\frac {\left (x -1\right )^{5}}{72}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (\frac {2}{135} (x-1)^5+\frac {17}{648} (x-1)^4+\frac {1}{27} (x-1)^3+\frac {1}{6} (x-1)^2+1\right )+c_2 \left (\frac {1}{72} (x-1)^5+\frac {1}{54} (x-1)^4+\frac {1}{18} (x-1)^3+x-1\right ) \]

✓ Sympy. Time used: 0.475 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{2} \left (\frac {17 \left (x - 1\right )^{4}}{648} + \frac {\left (x - 1\right )^{3}}{27} + \frac {\left (x - 1\right )^{2}}{6} + 1\right ) + C_{1} \left (x + \frac {\left (x - 1\right )^{4}}{54} + \frac {\left (x - 1\right )^{3}}{18} - 1\right ) + O\left (x^{6}\right ) \]