67.4.35 problem Problem 5(f)

Internal problem ID [14079]
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(f)
Date solved : Tuesday, January 28, 2025 at 06:13:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-7 y^{\prime }+6 y&=\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: 9.102 (sec). Leaf size: 23

dsolve([diff(y(t),t$2)-7*diff(y(t),t)+6*y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -1\right ) \left (-{\mathrm e}^{-6+6 t}+{\mathrm e}^{t -1}\right )}{5} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 102

DSolve[{D[y[t],{t,2}]-7*D[y[t],t]+6*y[t]==DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^t \left (-\int _1^t-\frac {\delta (K[1]-1)}{5 e}dK[1]+e^{5 t} \int _1^0\frac {\delta (K[2]-1)}{5 e^6}dK[2]-e^{5 t} \int _1^t\frac {\delta (K[2]-1)}{5 e^6}dK[2]+\int _1^0-\frac {\delta (K[1]-1)}{5 e}dK[1]\right ) \]