67.22.8 problem 31.7 (a)

Internal problem ID [16922]
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 (a)
Date solved : Thursday, October 02, 2025 at 01:40:23 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} 3 y+y^{\prime }&=\delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.106 (sec). Leaf size: 18

ode:=diff(y(t),t)+3*y(t) = Dirac(t-2); 
ic:=[y(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-3 t} \left ({\mathrm e}^{6} \operatorname {Heaviside}\left (t -2\right )+2\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 29

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

\begin{align*} y(t)&\to e^{-3 t} \left (\int _0^te^6 \delta (K[1]-2)dK[1]+2\right ) \end{align*}

✓ Sympy. Time used: 0.436 (sec). Leaf size: 46

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

\[ - \int \operatorname {Dirac}{\left (t - 2 \right )} e^{3 t}\, dt + 3 \int y{\left (t \right )} e^{3 t}\, dt = - \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{3 t}\, dt + 3 \int \limits ^{0} y{\left (t \right )} e^{3 t}\, dt \]