84.29.2 problem 18.5 (b)

Internal problem ID [22287]
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 (b)
Date solved : Thursday, October 02, 2025 at 08:37:07 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*} 2 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 62

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

\[ y = c_1 \left (x -2\right ) \left (1-\frac {1}{8} \left (x -2\right )+\frac {1}{64} \left (x -2\right )^{2}-\frac {7}{3072} \left (x -2\right )^{3}+\frac {91}{245760} \left (x -2\right )^{4}-\frac {637}{9830400} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+c_2 \left (\ln \left (x -2\right ) \left (-\frac {1}{4} \left (x -2\right )+\frac {1}{32} \left (x -2\right )^{2}-\frac {1}{256} \left (x -2\right )^{3}+\frac {7}{12288} \left (x -2\right )^{4}-\frac {91}{983040} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+\left (1-\frac {1}{64} \left (x -2\right )^{2}+\frac {1}{768} \left (x -2\right )^{3}-\frac {17}{147456} \left (x -2\right )^{4}+\frac {311}{29491200} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 114

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

\[ y(x)\to c_2 \left (\frac {91 (x-2)^5}{245760}-\frac {7 (x-2)^4}{3072}+\frac {1}{64} (x-2)^3-\frac {1}{8} (x-2)^2+x-2\right )+c_1 \left (\frac {-185 (x-2)^4+1344 (x-2)^3-11520 (x-2)^2+73728 (x-2)+147456}{147456}+\frac {\left (7 (x-2)^3-48 (x-2)^2+384 (x-2)-3072\right ) (x-2) \log (x-2)}{12288}\right ) \]

✓ Sympy. Time used: 0.407 (sec). Leaf size: 7

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_regular",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{1} + O\left (x^{6}\right ) \]