67.18.8 problem 27.1 (h)

Internal problem ID [16879]
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 (h)
Date solved : Thursday, October 02, 2025 at 01:40:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&={\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.094 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = exp(4*t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left ({\mathrm e}^{7 t}+119 \,{\mathrm e}^{t}-78\right ) {\mathrm e}^{-3 t}}{42} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==Exp[4*t]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{42} e^{-3 t} \left (119 e^t+e^{7 t}-78\right ) \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) - exp(4*t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {e^{4 t}}{42} + \frac {17 e^{- 2 t}}{6} - \frac {13 e^{- 3 t}}{7} \]