78.26.1 problem 4 (a)

Internal problem ID [18391]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 53. More about Convolutions. The Unit Step and Impulse Functions. Problems at page 481
Problem number : 4 (a)
Date solved : Monday, March 31, 2025 at 05:27:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=4 \,{\mathrm e}^{3 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.127 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = 4*exp(3*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {4 \,{\mathrm e}^{-2 t}}{5}+\frac {4 \cosh \left (3 t \right )}{5}-\frac {8 \sinh \left (3 t \right )}{15} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 26

ode=D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==4*Exp[3*t]; 
ic={y[0]==0,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {2}{15} e^{-3 t} \left (-6 e^t+e^{6 t}+5\right ) \]

✓ Sympy. Time used: 0.215 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) - 4*exp(3*t) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {2 e^{3 t}}{15} - \frac {4 e^{- 2 t}}{5} + \frac {2 e^{- 3 t}}{3} \]