52.8.8 problem 8

Internal problem ID [8372]
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 : 8
Date solved : Monday, January 27, 2025 at 03:49:43 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.693 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.119 (sec). Leaf size: 37

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

\[ y(t)\to \frac {1}{4} \left (\left (2 e^{2 t-4}-2\right ) \theta (t-2)-2 t+3 e^{2 t}-3\right ) \]