73.22.16 problem 31.7 (i)

Internal problem ID [15572]
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 (i)
Date solved : Monday, March 31, 2025 at 01:41:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=\delta \left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.100 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+9*y(t) = Dirac(t-4); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \operatorname {Heaviside}\left (t -4\right ) \left (t -4\right ) {\mathrm e}^{12-3 t} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 86

ode=D[y[t],{t,2}]+6*D[y[t],t]+9*y[t]==DiracDelta[t-4]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -e^{-3 t} \left (-\int _1^t-4 e^{12} \delta (K[1]-4)dK[1]+t \int _1^0e^{12} \delta (K[2]-4)dK[2]-t \int _1^te^{12} \delta (K[2]-4)dK[2]+\int _1^0-4 e^{12} \delta (K[1]-4)dK[1]\right ) \]

✓ Sympy. Time used: 1.248 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 4) + 9*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t \left (\int \operatorname {Dirac}{\left (t - 4 \right )} e^{3 t}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 4 \right )} e^{3 t}\, dt\right ) - \int t \operatorname {Dirac}{\left (t - 4 \right )} e^{3 t}\, dt + \int \limits ^{0} t \operatorname {Dirac}{\left (t - 4 \right )} e^{3 t}\, dt\right ) e^{- 3 t} \]