18.2.2 problem Problem 15.2(a)

Internal problem ID [3485]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.2(a)
Date solved : Tuesday, March 04, 2025 at 04:43:04 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} f^{\prime \prime }+2 f^{\prime }+5 f&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 21

ode:=diff(diff(f(t),t),t)+2*diff(f(t),t)+5*f(t) = 0; 
ic:=f(0) = 1, D(f)(0) = 0; 
dsolve([ode,ic],f(t), singsol=all);
 

\[ f = \frac {{\mathrm e}^{-t} \left (2 \cos \left (2 t \right )+\sin \left (2 t \right )\right )}{2} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 25

ode=D[ f[t],{t,2}]+2*D[ f[t],t]+5*f[t]==0; 
ic={f[0]==1,Derivative[1][f][0]==0}; 
DSolve[{ode,ic},f[t],t,IncludeSingularSolutions->True]
 

\[ f(t)\to \frac {1}{2} e^{-t} (\sin (2 t)+2 \cos (2 t)) \]

✓ Sympy. Time used: 0.173 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
f = Function("f") 
ode = Eq(5*f(t) + 2*Derivative(f(t), t) + Derivative(f(t), (t, 2)),0) 
ics = {f(0): 1, Subs(Derivative(f(t), t), t, 0): 0} 
dsolve(ode,func=f(t),ics=ics)
 

\[ f{\left (t \right )} = \left (\frac {\sin {\left (2 t \right )}}{2} + \cos {\left (2 t \right )}\right ) e^{- t} \]