Internal problem ID [11002]
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 14(b).
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }-16 y-32 \operatorname {Heaviside}\left (t \right )+32 \operatorname {Heaviside}\left (t -\pi \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 119
dsolve([diff(y(t),t$4)-16*y(t)=32*(Heaviside(t)-Heaviside(t-Pi)),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-2 t +2 \pi }}{2}-\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }}{2}+\left (2-\cos \left (2 t \right )\right ) \operatorname {Heaviside}\left (t -\pi \right )+\left (\cos \left (2 t \right )+\frac {{\mathrm e}^{-2 t}}{2}+\frac {{\mathrm e}^{2 t}}{2}-2\right ) \operatorname {Heaviside}\left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 39
DSolve[{y''''[t]-16*y[t]==32*(UnitStep[t]-UnitStep[t-Pi]),{y[0]==0,y'[0]==0,y''[0]==0,y'''[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \cos (2 t)+\cosh (2 t)-2 & 0\leq t\leq \pi \\ -2 \sinh (\pi ) \sinh (\pi -2 t) & t>\pi \\ \\ \\ \\ \\ \end{align*}