67.19.9 problem 28.9 (b)

Internal problem ID [16894]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 28. The inverse Laplace transform. Additional Exercises. page 509
Problem number : 28.9 (b)
Date solved : Thursday, October 02, 2025 at 01:40:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+40 y&=122 \,{\mathrm e}^{-3 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=8 \\ \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 29

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+40*y(t) = 122*exp(-3*t); 
ic:=[y(0) = 0, D(y)(0) = 8]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{2 t} \left (-2 \cos \left (6 t \right )+3 \sin \left (6 t \right )\right ) \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 35

ode=D[y[t],{t,2}]-4*D[y[t],t]+40*y[t]==122*Exp[-3*t]; 
ic={y[0]==0,Derivative[1][y][0] ==8}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-3 t} \left (3 e^{5 t} \sin (6 t)-2 e^{5 t} \cos (6 t)+2\right ) \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(40*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 122*exp(-3*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 8} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (3 \sin {\left (6 t \right )} - 2 \cos {\left (6 t \right )}\right ) e^{2 t} + 2 e^{- 3 t} \]