61.4.42 problem Problem 14(b)

Internal problem ID [15367]
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 : Thursday, October 02, 2025 at 10:12:13 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*}

✓ Maple. Time used: 0.207 (sec). Leaf size: 45

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-16*y(t) = 32*Heaviside(t)-32*Heaviside(t-Pi); 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 72

ode=D[y[t],{t,4}]-16*y[t]==32*(UnitStep[t]-UnitStep[t-Pi]); 
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]
 

\begin{align*} 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} \end{align*}

✓ Sympy. Time used: 1.676 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-16*y(t) - 32*Heaviside(t) + 32*Heaviside(t - pi) + 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)
 

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