76.21.5 problem 5

Internal problem ID [17769]
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 5.7 (Impulse Functions). Problems at page 350
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:02:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=\sin \left (t \right )+\delta \left (t -3 \pi \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 16.375 (sec). Leaf size: 54

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+3*y(t)=sin(t)+Dirac(t-3*Pi),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\sqrt {2}\, {\mathrm e}^{-t +3 \pi } \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (\sqrt {2}\, \left (t -3 \pi \right )\right )}{2}+\frac {{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )}{4}-\frac {\cos \left (t \right )}{4}+\frac {\sin \left (t \right )}{4} \]

✓ Solution by Mathematica

Time used: 1.598 (sec). Leaf size: 66

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+3*y[t]==Sin[t]+DiracDelta[t-3*Pi],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{4} \left (-2 \sqrt {2} e^{3 \pi -t} \theta (t-3 \pi ) \sin \left (\sqrt {2} (3 \pi -t)\right )+\sin (t)-\cos (t)+e^{-t} \cos \left (\sqrt {2} t\right )\right ) \]