72.20.3 problem 4

Internal problem ID [14969]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.4. page 608
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 07:25:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 10.077 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

Time used: 0.103 (sec). Leaf size: 110

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

\[ y(t)\to -e^{-t} \left (\sin (t) \int _1^0-2 e^2 \cos (2) \delta (K[1]-2)dK[1]-\sin (t) \int _1^t-2 e^2 \cos (2) \delta (K[1]-2)dK[1]+\cos (t) \int _1^02 e^2 \delta (K[2]-2) \sin (2)dK[2]-\cos (t) \int _1^t2 e^2 \delta (K[2]-2) \sin (2)dK[2]-2 (\sin (t)+\cos (t))\right ) \]