Internal problem ID [10981]
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 3(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y-39 \operatorname {Heaviside}\left (t \right )+507 \left (t -2\right ) \operatorname {Heaviside}\left (t -2\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 84
dsolve([diff(y(t),t$2)+4*diff(y(t),t)+13*y(t)=39*Heaviside(t)-507*(t-2)*Heaviside(t-2),y(0) = 3, D(y)(0) = 1],y(t), singsol=all)
\[ y \left (t \right ) = -12 \operatorname {Heaviside}\left (t -2\right ) \left (\left (\cos \left (6\right )+\frac {5 \sin \left (6\right )}{12}\right ) \cos \left (3 t \right )-\frac {5 \sin \left (3 t \right ) \left (\cos \left (6\right )-\frac {12 \sin \left (6\right )}{5}\right )}{12}\right ) {\mathrm e}^{-2 t +4}+3 \left (30-13 t \right ) \operatorname {Heaviside}\left (t -2\right )-3 \,{\mathrm e}^{-2 t} \left (\operatorname {Heaviside}\left (t \right )-1\right ) \cos \left (3 t \right )+\frac {\left (-6 \operatorname {Heaviside}\left (t \right )+7\right ) \sin \left (3 t \right ) {\mathrm e}^{-2 t}}{3}+3 \operatorname {Heaviside}\left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 103
DSolve[{y''[t]+4*y'[t]+13*y[t]==39*UnitStep[t]-507*(t-2)*UnitStep[t-2],{y[0]==3,y'[0]==1}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \frac {1}{3} e^{-2 t} \sin (3 t)+3 & 0\leq t\leq 2 \\ \frac {1}{3} e^{-2 t} (9 \cos (3 t)+7 \sin (3 t)) & t<0 \\ \frac {1}{3} e^{-2 t} \left (-9 e^{2 t} (13 t-31)-3 e^4 (12 \cos (6-3 t)+5 \sin (6-3 t))+\sin (3 t)\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}