6.12.14 problem 16

Internal problem ID [1868]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 05:20:43 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 48

Order:=6; 
ode:=diff(diff(y(x),x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=3);
 

\[ y = \left (1+\frac {\left (x -3\right )^{2}}{2}+\frac {\left (x -3\right )^{4}}{24}\right ) y \left (3\right )+\left (x -3+\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{5}}{120}\right ) y^{\prime }\left (3\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.0 (sec). Leaf size: 51

ode=D[y[x],{x,2}]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{24} (x-3)^4+\frac {1}{2} (x-3)^2+1\right )+c_2 \left (\frac {1}{120} (x-3)^5+\frac {1}{6} (x-3)^3+x-3\right ) \]

✓ Sympy. Time used: 0.176 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=3,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 3\right )^{4}}{24} + \frac {\left (x - 3\right )^{2}}{2} + 1\right ) + C_{1} \left (x + \frac {\left (x - 3\right )^{3}}{6} - 3\right ) + O\left (x^{6}\right ) \]