Internal problem ID [12350]
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]]
\[ \boxed {y^{\prime \prime \prime \prime }-16 y=-32 \operatorname {Heaviside}\left (t -\pi \right )+32 \operatorname {Heaviside}\left (t \right )} \] 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: 5.016 (sec). Leaf size: 40
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)
\[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\pi \right ) \cosh \left (2 t -2 \pi \right )+\left (-\cos \left (2 t \right )+2\right ) \operatorname {Heaviside}\left (t -\pi \right )+\cos \left (2 t \right )+\cosh \left (2 t \right )-2 \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 72
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]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-2 (t+\pi )} \left (-1+e^{2 \pi }\right ) \left (-e^{2 \pi }+e^{4 t}\right ) & t>\pi \\ \frac {1}{2} \left (2 \cos (2 t)+e^{-2 t}+e^{2 t}-4\right ) & 0\leq t\leq \pi \\ \end {array} \\ \end {array} \]