68.16.3 problem 3

Internal problem ID [17796]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 3
Date solved : Thursday, October 02, 2025 at 02:28:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{6}-\frac {25 \left (x -1\right )^{3}}{27}-\frac {2699 \left (x -1\right )^{4}}{648}-\frac {6404 \left (x -1\right )^{5}}{405}\right ) y \left (1\right )+\left (x -1+8 \left (x -1\right )^{2}+\frac {815 \left (x -1\right )^{3}}{18}+\frac {10991 \left (x -1\right )^{4}}{54}+\frac {834547 \left (x -1\right )^{5}}{1080}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (-\frac {6404}{405} (x-1)^5-\frac {2699}{648} (x-1)^4-\frac {25}{27} (x-1)^3-\frac {1}{6} (x-1)^2+1\right )+c_2 \left (\frac {834547 (x-1)^5}{1080}+\frac {10991}{54} (x-1)^4+\frac {815}{18} (x-1)^3+8 (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.330 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((16*x + 32)*Derivative(y(x), x) + (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)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {10991 \left (x - 1\right )^{4}}{54} + \frac {815 \left (x - 1\right )^{3}}{18} + 8 \left (x - 1\right )^{2} - 1\right ) + C_{1} \left (- \frac {2699 \left (x - 1\right )^{4}}{648} - \frac {25 \left (x - 1\right )^{3}}{27} - \frac {\left (x - 1\right )^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]