76.21.11 problem 11

Internal problem ID [17775]
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 : 11
Date solved : Tuesday, January 28, 2025 at 11:02:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\cos \left (t \right )+\delta \left (t -\frac {\pi }{2}\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: 13.441 (sec). Leaf size: 46

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

\[ y = -\cos \left (t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{\frac {\pi }{2}-t}+\frac {\left (-\cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-t}}{5}+\frac {\cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5} \]

✓ Solution by Mathematica

Time used: 0.157 (sec). Leaf size: 52

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

\[ y(t)\to \frac {1}{5} e^{-t} \left (-5 e^{\pi /2} \theta (2 t-\pi ) \cos (t)+\left (2 e^t-3\right ) \sin (t)+\left (e^t-1\right ) \cos (t)\right ) \]