90.29.2 problem 10

Internal problem ID [25434]
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 : 10
Date solved : Friday, October 03, 2025 at 12:01:30 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+4 y&=\delta \left (-3+t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 18

ode:=diff(y(t),t)+4*y(t) = Dirac(t-3); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-4 t} \left ({\mathrm e}^{12} \operatorname {Heaviside}\left (t -3\right )+1\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 21

ode=D[y[t],t]+4*y[t]==DiracDelta[t-3]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-4 t} \left (e^{12} \theta (t-3)+1\right ) \end{align*}

✓ Sympy. Time used: 0.446 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 3) + 4*y(t) + Derivative(y(t), t),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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