67.4.42 problem Problem 14(b)

Internal problem ID [14086]
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)
Date solved : Tuesday, January 28, 2025 at 06:14:01 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=32 \operatorname {Heaviside}\left (t \right )-32 \operatorname {Heaviside}\left (t -\pi \right ) \end{align*}

Using Laplace method 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: 9.885 (sec). Leaf size: 45

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)
 

\[ y = -\operatorname {Heaviside}\left (t -\pi \right ) \cosh \left (-2 \pi +2 t \right )+\left (2-\cos \left (2 t \right )\right ) \operatorname {Heaviside}\left (t -\pi \right )+\cos \left (2 t \right )+\cosh \left (2 t \right )-2 \]

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 72

DSolve[{D[y[t],{t,4}]-16*y[t]==32*(UnitStep[t]-UnitStep[t-Pi]),{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-2 (t+\pi )} \left (-1+e^{2 \pi }\right ) \left (-e^{2 \pi }+e^{4 t}\right ) & t>\pi \\ \frac {1}{2} \left (2 \cos (2 t)+e^{-2 t}+e^{2 t}-4\right ) & 0\leq t\leq \pi \\ \end {array} \\ \end {array} \]