64.20.7 problem 7

Internal problem ID [13579]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 7
Date solved : Monday, March 31, 2025 at 08:01:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=18 \,{\mathrm e}^{-t} \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.180 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-2*y(t) = 18*exp(-t)*sin(3*t); 
ic:=y(0) = 0, D(y)(0) = 3; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 30

ode=D[y[t],{t,2}]-D[y[t],t]-2*y[t]==18*Exp[-t]*Sin[3*t]; 
ic={y[0]==0,Derivative[1][y][0]==3}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.309 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) - Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 18*exp(-t)*sin(3*t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

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