44.24.1 problem 1(a)

Internal problem ID [9453]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 7. Laplace Transforms. Section 7.5 The Unit Step and Impulse Functions. Page 303
Problem number : 1(a)
Date solved : Tuesday, September 30, 2025 at 06:18:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=5 \,{\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: 0.114 (sec). Leaf size: 21

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

\[ y = \cosh \left (3 t \right )-\frac {2 \sinh \left (3 t \right )}{3}-{\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to \frac {1}{6} e^{-3 t} \left (-6 e^t+e^{6 t}+5\right ) \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) - 5*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)
 

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