90.28.3 problem 3

Internal problem ID [25425]
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 437
Problem number : 3
Date solved : Friday, October 03, 2025 at 12:01:25 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 18

ode:=diff(y(t),t)-4*y(t) = Dirac(t-4); 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{4 t} \left (2+{\mathrm e}^{-16} \operatorname {Heaviside}\left (t -4\right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 23

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

\begin{align*} y(t)&\to e^{4 (t-4)} \left (\theta (t-4)+2 e^{16}\right ) \end{align*}

✓ Sympy. Time used: 0.408 (sec). Leaf size: 49

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

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