76.21.12 problem 12

Internal problem ID [17705]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 12
Date solved : Monday, March 31, 2025 at 04:25:26 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y&=\delta \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: 0.123 (sec). Leaf size: 21

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-y(t) = Dirac(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');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -1\right ) \left (\sin \left (t -1\right )-\sinh \left (t -1\right )\right )}{2} \]

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 44

ode=D[y[t],{t,4}]-y[t]==DiracDelta[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]
 

\[ y(t)\to \frac {1}{4} e^{-t-1} \theta (t-1) \left (e^{2 t}+2 e^{t+1} \sin (1-t)-e^2\right ) \]

✓ Sympy. Time used: 1.480 (sec). Leaf size: 110

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) - y(t) + 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)
 

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{4} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{- t}\, dt}{4}\right ) e^{t} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt}{4} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{t}\, dt}{4}\right ) e^{- t} + \left (\frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (t \right )}\, dt}{2} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (t \right )}\, dt}{2}\right ) \cos {\left (t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (t \right )}\, dt}{2} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (t \right )}\, dt}{2}\right ) \sin {\left (t \right )} \]