52.8.7 problem 7

Internal problem ID [8371]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. EXERCISES 7.5. Page 315
Problem number : 7
Date solved : Monday, January 27, 2025 at 03:49:42 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }&=\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 )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.692 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 37

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

\[ y(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^{2 t}-e^2\right ) \theta (t-1)+e^{2 t}-1\right ) \]