11.5.7 problem 7

Internal problem ID [1512]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.5, The Laplace Transform. Impulse functions. page 273
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:58:31 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: 0.459 (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)
 

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

✓ Solution by Mathematica

Time used: 0.160 (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]
 

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