7.13.12 problem 12

Internal problem ID [411]
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 : 12
Date solved : Saturday, March 29, 2025 at 04:53:03 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

Order:=6; 
ode:=diff(diff(y(x),x),x) = 4*y(x); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+2 x^{2}+\frac {2}{3} x^{4}\right ) y \left (0\right )+\left (x +\frac {2}{3} x^{3}+\frac {2}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {2 x^5}{15}+\frac {2 x^3}{3}+x\right )+c_1 \left (\frac {2 x^4}{3}+2 x^2+1\right ) \]

✓ Sympy. Time used: 0.681 (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 = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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