76.18.8 problem 19

Internal problem ID [17648]
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 : 19
Date solved : Monday, March 31, 2025 at 04:23:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=t \cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.170 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = t*cos(2*t); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\cos \left (2 t \right ) \left (-2+17 t +291 \,{\mathrm e}^{-t}\right )}{289}+\frac {4 \left (t +\frac {213 \,{\mathrm e}^{-t}}{68}-\frac {19}{17}\right ) \sin \left (2 t \right )}{17} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 47

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

\[ y(t)\to \frac {1}{289} e^{-t} \left (\left (e^t (68 t-76)+213\right ) \sin (2 t)+\left (e^t (17 t-2)+291\right ) \cos (2 t)\right ) \]

✓ Sympy. Time used: 0.337 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*cos(2*t) + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {4 t \sin {\left (2 t \right )}}{17} + \frac {t \cos {\left (2 t \right )}}{17} + \left (\frac {213 \sin {\left (2 t \right )}}{289} + \frac {291 \cos {\left (2 t \right )}}{289}\right ) e^{- t} - \frac {76 \sin {\left (2 t \right )}}{289} - \frac {2 \cos {\left (2 t \right )}}{289} \]