73.22.14 problem 31.7 (g)

Internal problem ID [15641]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (g)
Date solved : Tuesday, January 28, 2025 at 08:03:30 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }-12 y&=\delta \left (t \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: 9.331 (sec). Leaf size: 14

dsolve([diff(y(t),t$2)+4*diff(y(t),t)-12*y(t)=Dirac(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-2 t} \sinh \left (4 t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 84

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]-12*y[t]==DiracDelta[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-6 t} \left (-\int _1^t-\frac {\delta (K[1])}{8}dK[1]+e^{8 t} \int _1^0\frac {\delta (K[2])}{8}dK[2]-e^{8 t} \int _1^t\frac {\delta (K[2])}{8}dK[2]+\int _1^0-\frac {\delta (K[1])}{8}dK[1]\right ) \]