Internal problem ID [6707]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=\delta \left (-\pi +t \right )+\delta \left (t -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 56
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=Dirac(t-Pi)+Dirac(t-3*Pi),y(0) = 1, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\sin \left (3 t \right ) \operatorname {Heaviside}\left (t -3 \pi \right ) {\mathrm e}^{6 \pi -2 t}}{3}-\frac {\sin \left (3 t \right ) \operatorname {Heaviside}\left (-\pi +t \right ) {\mathrm e}^{-2 t +2 \pi }}{3}+{\mathrm e}^{-2 t} \left (\cos \left (3 t \right )+\frac {2 \sin \left (3 t \right )}{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.084 (sec). Leaf size: 59
DSolve[{y''[t]+4*y'[t]+13*y[t]==DiracDelta[t-Pi]+DiracDelta[t-3*Pi],{y[0]==1,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to -\frac {1}{3} e^{-2 t} \left (e^{6 \pi } \theta (t-3 \pi ) \sin (3 t)+e^{2 \pi } \theta (t-\pi ) \sin (3 t)-2 \sin (3 t)-3 \cos (3 t)\right ) \]