70.21.12 problem 12

Internal problem ID [19062]
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 : Thursday, October 02, 2025 at 03:37:38 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,op(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: 0.091 (sec). Leaf size: 198

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]
 

\begin{align*} y(t)&\to -e^{-t} \left (e^{2 t} \int _1^0\frac {\delta (K[1]-1)}{4 e}dK[1]-e^{2 t} \int _1^t\frac {\delta (K[1]-1)}{4 e}dK[1]-\int _1^t-\frac {1}{4} e \delta (K[3]-1)dK[3]+e^t \sin (t) \int _1^0-\frac {1}{2} \cos (1) \delta (K[4]-1)dK[4]-e^t \sin (t) \int _1^t-\frac {1}{2} \cos (1) \delta (K[4]-1)dK[4]+e^t \cos (t) \int _1^0\frac {1}{2} \delta (K[2]-1) \sin (1)dK[2]-e^t \cos (t) \int _1^t\frac {1}{2} \delta (K[2]-1) \sin (1)dK[2]+\int _1^0-\frac {1}{4} e \delta (K[3]-1)dK[3]\right ) \end{align*}

✓ Sympy. Time used: 0.882 (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)
 

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