54.3.2 problem 2

Internal problem ID [8558]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 2
Date solved : Wednesday, March 05, 2025 at 06:09:36 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-9 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

Order:=8; 
ode:=diff(diff(y(x),x),x)-9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\frac {81}{80} x^{6}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{3}+\frac {27}{40} x^{5}+\frac {81}{560} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

ode=D[y[x],{x,2}]-9*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {81 x^7}{560}+\frac {27 x^5}{40}+\frac {3 x^3}{2}+x\right )+c_1 \left (\frac {81 x^6}{80}+\frac {27 x^4}{8}+\frac {9 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.743 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {81 x^{6}}{80} + \frac {27 x^{4}}{8} + \frac {9 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {27 x^{4}}{40} + \frac {3 x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]