67.18.5 problem 27.1 (e)

Internal problem ID [16876]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (e)
Date solved : Thursday, October 02, 2025 at 01:40:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=20 \,{\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=12 \\ \end{align*}

✓ Maple. Time used: 0.111 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+4*y(t) = 20*exp(4*t); 
ic:=[y(0) = 3, D(y)(0) = 12]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{4 t}+2 \cos \left (2 t \right )+4 \sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+4*y[t]==20*Exp[4*t]; 
ic={y[0]==3,Derivative[1][y][0] ==12}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{4 t}+4 \sin (2 t)+2 \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 20*exp(4*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 12} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = e^{4 t} + 4 \sin {\left (2 t \right )} + 2 \cos {\left (2 t \right )} \]