70.19.6 problem 6

Internal problem ID [19016]
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 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 6
Date solved : Thursday, October 02, 2025 at 03:37:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+29 y&={\mathrm e}^{-2 t} \sin \left (5 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.198 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+29*y(t) = exp(-2*t)*sin(5*t); 
ic:=[y(0) = 5, D(y)(0) = -2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-2 t} \left (81 \sin \left (5 t \right )-5 \cos \left (5 t \right ) \left (-50+t \right )\right )}{50} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 119

ode=D[y[t],{t,2}]+4*D[y[t],t]+29*y[t]==Exp[-2*t]*Sin[5*t]; 
ic={y[0]==5,Derivative[1][y][0] == -2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{5} e^{-2 t} \left (-5 \sin (5 t) \int _1^0\frac {1}{10} \sin (10 K[1])dK[1]+5 \sin (5 t) \int _1^t\frac {1}{10} \sin (10 K[1])dK[1]-5 \cos (5 t) \int _1^0-\frac {1}{5} \sin ^2(5 K[2])dK[2]+5 \cos (5 t) \int _1^t-\frac {1}{5} \sin ^2(5 K[2])dK[2]+8 \sin (5 t)+25 \cos (5 t)\right ) \end{align*}

✓ Sympy. Time used: 0.274 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(29*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-2*t)*sin(5*t),0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (5 - \frac {t}{10}\right ) \cos {\left (5 t \right )} + \frac {81 \sin {\left (5 t \right )}}{50}\right ) e^{- 2 t} \]