73.22.19 problem 31.7 (L)

Internal problem ID [15575]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (L)
Date solved : Monday, March 31, 2025 at 01:41:33 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=\delta \left (t \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: 0.176 (sec). Leaf size: 17

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-16*y(t) = Dirac(t); 
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');
 

\[ y = -\frac {\sin \left (2 t \right )}{16}+\frac {\sinh \left (2 t \right )}{16} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 193

ode=D[y[t],{t,4}]-16*y[t]==DiracDelta[t]; 
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]
 

\[ y(t)\to \frac {1}{128} e^{-2 t} \left (-128 e^{4 t} \int _1^0\frac {\delta (K[1])}{32}dK[1]+128 e^{4 t} \int _1^t\frac {\delta (K[1])}{32}dK[1]+128 \int _1^t-\frac {\delta (K[2])}{32}dK[2]-128 e^{2 t} \sin (2 t) \int _1^0-\frac {\delta (K[3])}{16}dK[3]+128 e^{2 t} \sin (2 t) \int _1^t-\frac {\delta (K[3])}{16}dK[3]-128 \int _1^0-\frac {\delta (K[2])}{32}dK[2]-\delta ''(0)+\delta (0) e^{4 t}-2 \delta (0) e^{2 t} \cos (2 t)+\delta (0)+e^{4 t} \delta ''(0)-2 e^{2 t} \delta ''(0) \sin (2 t)\right ) \]

✓ Sympy. Time used: 2.255 (sec). Leaf size: 117

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t) - 16*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)
 

\[ y{\left (t \right )} = \left (\frac {\int \operatorname {Dirac}{\left (t \right )} e^{- 2 t}\, dt}{32} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} e^{- 2 t}\, dt}{32}\right ) e^{2 t} + \left (- \frac {\int \operatorname {Dirac}{\left (t \right )} e^{2 t}\, dt}{32} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} e^{2 t}\, dt}{32}\right ) e^{- 2 t} + \left (\frac {\int \operatorname {Dirac}{\left (t \right )} \sin {\left (2 t \right )}\, dt}{16} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \sin {\left (2 t \right )}\, dt}{16}\right ) \cos {\left (2 t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t \right )} \cos {\left (2 t \right )}\, dt}{16} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t \right )} \cos {\left (2 t \right )}\, dt}{16}\right ) \sin {\left (2 t \right )} \]