72.20.1 problem 2

Internal problem ID [14967]
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 : 2
Date solved : Tuesday, January 28, 2025 at 07:25:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y&=5 \delta \left (t -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: 10.763 (sec). Leaf size: 21

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

\[ y = \frac {5 \sqrt {3}\, \operatorname {Heaviside}\left (t -2\right ) \sin \left (\sqrt {3}\, \left (t -2\right )\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 160

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

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