67.4.41 problem Problem 14(a)

Internal problem ID [14006]
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 : Wednesday, March 05, 2025 at 10:25:22 PM
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*}

✓ Maple. Time used: 9.935 (sec). Leaf size: 51

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-5*diff(diff(y(t),t),t)+4*y(t) = 12*Heaviside(t)-12*Heaviside(t-1); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 1.581 (sec). Leaf size: 109

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 12*Heaviside(t) + 12*Heaviside(t - 1) - 5*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- 2 \theta \left (t\right ) + 2 e \theta \left (t - 1\right )\right ) e^{- t} + \left (- 2 \theta \left (t\right ) + \frac {2 \theta \left (t - 1\right )}{e}\right ) e^{t} + \left (\frac {\theta \left (t\right )}{2} - \frac {\theta \left (t - 1\right )}{2 e^{2}}\right ) e^{2 t} + \left (\frac {\theta \left (t\right )}{2} - \frac {e^{2} \theta \left (t - 1\right )}{2}\right ) e^{- 2 t} + 3 \theta \left (t\right ) - 3 \theta \left (t - 1\right ) \]