90.29.4 problem 14

Internal problem ID [25436]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 449
Problem number : 14
Date solved : Friday, October 03, 2025 at 12:01:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=\delta \left (-3+t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)-6*diff(y(t),t)+9*y(t) = Dirac(t-3); 
ic:=[y(0) = -1, D(y)(0) = -3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{3 t} \left (\operatorname {Heaviside}\left (t -3\right ) \left (t -3\right ) {\mathrm e}^{-9}-1\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 27

ode=D[y[t],{t,2}]-6*D[y[t],t]+9*y[t]==DiracDelta[t-3]; 
ic={y[0]==-1,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{3 t-9} \left ((t-3) \theta (t-3)-e^9\right ) \end{align*}

✓ Sympy. Time used: 0.834 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3) + 9*y(t) - 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): -3} 
dsolve(ode,func=y(t),ics=ics)
 

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