7.13.20 problem 20

Internal problem ID [419]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 20
Date solved : Saturday, March 29, 2025 at 04:53:12 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

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

Order:=6; 
ode:=diff(diff(y(x),x),x)-4*y(x) = 0; 
ic:=y(0) = 2, D(y)(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 17

ode=D[y[x],{x,2}]-4*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {4 x^4}{3}+4 x^2+2 \]

✓ Sympy. Time used: 0.688 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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