Internal problem ID [10982]
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(h).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+4 y-3 \operatorname {Heaviside}\left (t \right )+3 \operatorname {Heaviside}\left (t -4\right )-\left (2 t -5\right ) \operatorname {Heaviside}\left (t -4\right )=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {3}{4}}, y^{\prime }\left (0\right ) = 2\right ] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 48
dsolve([diff(y(t),t$2)+4*y(t)=3*(Heaviside(t)-Heaviside(t-4))+(2*t-5)*Heaviside(t-4),y(0) = 3/4, D(y)(0) = 2],y(t), singsol=all)
\[ y \left (t \right ) = \sin \left (2 t \right )+\frac {3 \cos \left (2 t \right )}{4}-\frac {\operatorname {Heaviside}\left (t -4\right ) \sin \left (2 t -8\right )}{4}+\frac {\operatorname {Heaviside}\left (t -4\right ) t}{2}-2 \operatorname {Heaviside}\left (t -4\right )-\frac {3 \operatorname {Heaviside}\left (t \right ) \cos \left (2 t \right )}{4}+\frac {3 \operatorname {Heaviside}\left (t \right )}{4} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 59
DSolve[{y''[t]+4*y[t]==3*(UnitStep[t]-UnitStep[t-4])+(2*t-5)*UnitStep[t-4],{y[0]==3/4,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \sin (2 t)+\frac {3}{4} & 0\leq t\leq 4 \\ \frac {3}{4} \cos (2 t)+\sin (2 t) & t<0 \\ \frac {1}{4} (2 t+\sin (8-2 t)-5)+\sin (2 t) & \text {True} \\ \\ \\ \\ \\ \end{align*}