Internal problem ID [10983]
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(i).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+5 y-25 t \left (\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 85
dsolve([4*diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=25*t*(Heaviside(t)-Heaviside(t-Pi/2)),y(0) = 2, D(y)(0) = 2],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {5 \left (\left (\pi +\frac {12}{5}\right ) \cos \left (t \right )-2 \left (\pi -\frac {8}{5}\right ) \sin \left (t \right )\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{4}+\left (4-5 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )+\left (\left (4 \cos \left (t \right )-3 \sin \left (t \right )\right ) \operatorname {Heaviside}\left (t \right )+2 \cos \left (t \right )+3 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}}+\operatorname {Heaviside}\left (t \right ) \left (-4+5 t \right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 97
DSolve[{4*y''[t]+4*y'[t]+5*y[t]==25*t*(UnitStep[t]-UnitStep[t-Pi/2]),{y[0]==2,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} 5 t+6 e^{-t/2} \cos (t)-4 & t\geq 0\land 2 t\leq \pi \\ e^{-t/2} (2 \cos (t)+3 \sin (t)) & t<0 \\ \frac {1}{4} e^{-t/2} \left (24 \cos (t)-e^{\pi /4} ((12+5 \pi ) \cos (t)+2 (8-5 \pi ) \sin (t))\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}