73.22.18 problem 31.7 (k)

Internal problem ID [15574]
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 (k)
Date solved : Monday, March 31, 2025 at 01:41:31 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+9 y^{\prime }&=\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 \end{align*}

✓ Maple. Time used: 0.122 (sec). Leaf size: 18

ode:=diff(diff(diff(y(t),t),t),t)+9*diff(y(t),t) = Dirac(t-1); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -1\right ) \left (-1+\cos \left (-3+3 t \right )\right )}{9} \]

✓ Mathematica. Time used: 60.035 (sec). Leaf size: 224

ode=D[ y[t],{t,3}]+9*D[y[t],t]==DiracDelta[t-1]; 
ic={y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t\left (-\sin (3 K[3]) \int _1^0\frac {1}{3} \cos (3) \delta (K[2]-1)dK[2]+\sin (3 K[3]) \int _1^{K[3]}\frac {1}{3} \cos (3) \delta (K[2]-1)dK[2]-\cos (3 K[3]) \int _1^0-\frac {1}{3} \delta (K[1]-1) \sin (3)dK[1]+\cos (3 K[3]) \int _1^{K[3]}-\frac {1}{3} \delta (K[1]-1) \sin (3)dK[1]\right )dK[3]-\int _1^0\left (-\sin (3 K[3]) \int _1^0\frac {1}{3} \cos (3) \delta (K[2]-1)dK[2]+\sin (3 K[3]) \int _1^{K[3]}\frac {1}{3} \cos (3) \delta (K[2]-1)dK[2]-\cos (3 K[3]) \int _1^0-\frac {1}{3} \delta (K[1]-1) \sin (3)dK[1]+\cos (3 K[3]) \int _1^{K[3]}-\frac {1}{3} \delta (K[1]-1) \sin (3)dK[1]\right )dK[3] \]

✓ Sympy. Time used: 1.174 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 9*Derivative(y(t), t) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (3 t \right )}\, dt}{9} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \sin {\left (3 t \right )}\, dt}{9}\right ) \sin {\left (3 t \right )} + \left (- \frac {\int \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (3 t \right )}\, dt}{9} + \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} \cos {\left (3 t \right )}\, dt}{9}\right ) \cos {\left (3 t \right )} + \frac {\int \operatorname {Dirac}{\left (t - 1 \right )}\, dt}{9} - \frac {\int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )}\, dt}{9} \]