67.4.31 problem Problem 5(b)

Internal problem ID [14075]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(b)
Date solved : Tuesday, January 28, 2025 at 06:13:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=3 \delta \left (t -1\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: 9.046 (sec). Leaf size: 20

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+2*y(t)=3*Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = 3 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t} \sin \left (t -1\right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 95

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

\[ y(t)\to -e^{-t} \left (\sin (t) \int _1^03 e \cos (1) \delta (K[1]-1)dK[1]-\sin (t) \int _1^t3 e \cos (1) \delta (K[1]-1)dK[1]+\cos (t) \int _1^0-3 e \delta (K[2]-1) \sin (1)dK[2]-\cos (t) \int _1^t-3 e \delta (K[2]-1) \sin (1)dK[2]\right ) \]