78.3.8 problem 2.c

Internal problem ID [21004]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 2.c
Date solved : Thursday, October 02, 2025 at 07:01:19 PM
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*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

Order:=5; 
ode:=diff(diff(y(x),x),x)+9*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1-\frac {9}{2} x^{2}+\frac {27}{8} x^{4}+\operatorname {O}\left (x^{5}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

ode=D[y[x],{x,2}]+9*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

\[ y(x)\to \frac {27 x^4}{8}-\frac {9 x^2}{2}+1 \]

✓ Sympy. Time used: 0.174 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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