Internal problem ID [6703]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 1.984 (sec). Leaf size: 28
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)
\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}}{2}+\frac {\operatorname {Heaviside}\left (t -1\right )}{2}+\frac {1}{2}-\frac {{\mathrm e}^{-2 t}}{2} \]
✓ Solution by Mathematica
Time used: 0.144 (sec). Leaf size: 37
DSolve[{y''[t]+2*y'[t]==DiracDelta[t-1],{y[0]==0,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^{2 t}-e^2\right ) \theta (t-1)+e^{2 t}-1\right ) \]