72.20.2 problem 3

Internal problem ID [14968]
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 : 3
Date solved : Tuesday, January 28, 2025 at 07:25:48 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=\delta \left (t -3\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 11.059 (sec). Leaf size: 37

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+5*y(t)=Dirac(t-3),y(0) = 1, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = {\mathrm e}^{-t} \left (\sin \left (2 t \right )+\cos \left (2 t \right )\right )+\frac {{\mathrm e}^{-t +3} \operatorname {Heaviside}\left (t -3\right ) \sin \left (2 t -6\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.072 (sec). Leaf size: 131

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==DiracDelta[t-3],{y[0]==1,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-t} \left (\cos (2 t) \int _1^0-e^3 \cos (3) \delta (K[2]-3) \sin (3)dK[2]-\cos (2 t) \int _1^t-e^3 \cos (3) \delta (K[2]-3) \sin (3)dK[2]+\sin (2 t) \int _1^0\frac {1}{2} e^3 \cos (6) \delta (K[1]-3)dK[1]-\sin (2 t) \int _1^t\frac {1}{2} e^3 \cos (6) \delta (K[1]-3)dK[1]-\sin (2 t)-\cos (2 t)\right ) \]