12.2.5 problem Problem 15.5(a)

Internal problem ID [3488]
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 15.5(a)
Date solved : Tuesday, September 30, 2025 at 06:40:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} f^{\prime \prime }+8 f^{\prime }+12 f&=12 \,{\mathrm e}^{-4 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 14

ode:=diff(diff(f(t),t),t)+8*diff(f(t),t)+12*f(t) = 12*exp(-4*t); 
ic:=[f(0) = 0, D(f)(0) = 0]; 
dsolve([ode,op(ic)],f(t), singsol=all);
 

\[ f = 6 \sinh \left (t \right )^{2} {\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 23

ode=D[ f[t],{t,2}]+8*D[ f[t],t]+12*f[t]==12*Exp[-4*t]; 
ic={f[0]==0,Derivative[1][f][0]==0}; 
DSolve[{ode,ic},f[t],t,IncludeSingularSolutions->True]
 

\begin{align*} f(t)&\to \frac {3}{2} e^{-6 t} \left (e^{2 t}-1\right )^2 \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
f = Function("f") 
ode = Eq(12*f(t) + 8*Derivative(f(t), t) + Derivative(f(t), (t, 2)) - 12*exp(-4*t),0) 
ics = {f(0): 0, Subs(Derivative(f(t), t), t, 0): 0} 
dsolve(ode,func=f(t),ics=ics)
 

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