72.20.4 problem 5

Internal problem ID [14970]
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 : 5
Date solved : Tuesday, January 28, 2025 at 07:25:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+3 y&=\delta \left (t -1\right )-3 \delta \left (t -4\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: 12.681 (sec). Leaf size: 49

dsolve([diff(y(t),t$2)+2*diff(y(t),t)+3*y(t)=Dirac(t-1)-3*Dirac(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\sqrt {2}\, \left (-3 \operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4} \sin \left (\sqrt {2}\, \left (t -4\right )\right )+\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t} \sin \left (\sqrt {2}\, \left (t -1\right )\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.357 (sec). Leaf size: 217

DSolve[{D[y[t],{t,2}]+2*D[y[t],t]+3*y[t]==DiracDelta[t-1]-3*DiracDelta[t-4],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-t} \left (\sin \left (\sqrt {2} t\right ) \int _1^0-\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (3 \delta (K[1]-4)-\delta (K[1]-1))}{\sqrt {2}}dK[1]-\sin \left (\sqrt {2} t\right ) \int _1^t-\frac {e^{K[1]} \cos \left (\sqrt {2} K[1]\right ) (3 \delta (K[1]-4)-\delta (K[1]-1))}{\sqrt {2}}dK[1]+\cos \left (\sqrt {2} t\right ) \int _1^0\frac {e^{K[2]} (3 \delta (K[2]-4)-\delta (K[2]-1)) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]-\cos \left (\sqrt {2} t\right ) \int _1^t\frac {e^{K[2]} (3 \delta (K[2]-4)-\delta (K[2]-1)) \sin \left (\sqrt {2} K[2]\right )}{\sqrt {2}}dK[2]\right ) \]