12.2.3 problem Problem 15.2(b)

Internal problem ID [3486]
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(b)
Date solved : Tuesday, September 30, 2025 at 06:40:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} f^{\prime \prime }+2 f^{\prime }+5 f&={\mathrm e}^{-t} \cos \left (3 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 28

ode:=diff(diff(f(t),t),t)+2*diff(f(t),t)+5*f(t) = exp(-t)*cos(3*t); 
ic:=[f(0) = 0, D(f)(0) = 0]; 
dsolve([ode,op(ic)],f(t), singsol=all);
 

\[ f = -\frac {\left (4 \cos \left (t \right )^{3}-2 \cos \left (t \right )^{2}-3 \cos \left (t \right )+1\right ) {\mathrm e}^{-t}}{5} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 34

ode=D[ f[t],{t,2}]+2*D[ f[t],t]+5*f[t]==Exp[-t]*Cos[3*t]; 
ic={f[0]==0,Derivative[1][f][0]==0}; 
DSolve[{ode,ic},f[t],t,IncludeSingularSolutions->True]
 

\begin{align*} f(t)&\to \frac {2}{5} e^{-t} \sin ^2\left (\frac {t}{2}\right ) (2 \cos (t)+2 \cos (2 t)+1) \end{align*}

✓ Sympy. Time used: 0.208 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
f = Function("f") 
ode = Eq(5*f(t) + 2*Derivative(f(t), t) + Derivative(f(t), (t, 2)) - exp(-t)*cos(3*t),0) 
ics = {f(0): 0, Subs(Derivative(f(t), t), t, 0): 0} 
dsolve(ode,func=f(t),ics=ics)
 

\[ f{\left (t \right )} = \left (\frac {\cos {\left (2 t \right )}}{5} - \frac {\cos {\left (3 t \right )}}{5}\right ) e^{- t} \]