76.18.7 problem 18

Internal problem ID [17647]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 18
Date solved : Monday, March 31, 2025 at 04:23:33 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=3 \,{\mathrm e}^{-2 t} \sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.192 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*exp(-2*t)*sin(2*t); 
ic:=y(0) = 2, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\cos \left (2 t \right ) \left (17+3 \,{\mathrm e}^{-2 t}\right )}{10}-\frac {\sin \left (2 t \right ) \left (7-3 \,{\mathrm e}^{-2 t}\right )}{20} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 44

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

\[ y(t)\to \frac {1}{20} e^{-2 t} \left (\left (34 e^{2 t}+6\right ) \cos (2 t)-\left (7 e^{2 t}-3\right ) \sin (2 t)\right ) \]

✓ Sympy. Time used: 0.181 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + Derivative(y(t), (t, 2)) - 3*exp(-2*t)*sin(2*t),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {7}{20} + \frac {3 e^{- 2 t}}{20}\right ) \sin {\left (2 t \right )} + \left (\frac {17}{10} + \frac {3 e^{- 2 t}}{10}\right ) \cos {\left (2 t \right )} \]