67.4.41 problem Problem 14(a)

Internal problem ID [14085]
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(a)
Date solved : Tuesday, January 28, 2025 at 06:13:51 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y&=12 \operatorname {Heaviside}\left (t \right )-12 \operatorname {Heaviside}\left (t -1\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: 9.935 (sec). Leaf size: 51

dsolve([diff(y(t),t$4)-5*diff(y(t),t$2)+4*y(t)=12*(Heaviside(t)-Heaviside(t-1)),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = 2 \left (\cosh \left (t \right )-1\right )^{2}+\frac {\operatorname {Heaviside}\left (t -1\right ) \left (4 \,{\mathrm e}^{1-t}-{\mathrm e}^{2-2 t}-{\mathrm e}^{2 t -2}-6+4 \,{\mathrm e}^{t -1}\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 88

DSolve[{D[y[t],{t,4}]-5*D[y[t],{t,2}]+4*y[t]==12*(UnitStep[t]-UnitStep[t-1]),{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} e^{-2 t} \left (-1+e^t\right )^4 & 0\leq t\leq 1 \\ \frac {1}{2} (-1+e) e^{-2 (t+1)} \left (-e^2-e^3+e^{4 t}+4 e^{t+2}-4 e^{3 t+1}+e^{4 t+1}\right ) & t>1 \\ \end {array} \\ \end {array} \]