73.22.9 problem 31.7 (b)

Internal problem ID [15557]
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 (b)
Date solved : Thursday, March 13, 2025 at 06:11:27 AM
CAS classification : [[_2nd_order, _missing_y]]

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

Using Laplace method

✓ Maple. Time used: 9.337 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t) = Dirac(t); 
dsolve(ode,y(t),method='laplace');
 

\[ y = \frac {1}{3}+\frac {y^{\prime }\left (0\right )}{3}+y \left (0\right )-\frac {{\mathrm e}^{-3 t} \left (1+y^{\prime }\left (0\right )\right )}{3} \]

✓ Mathematica. Time used: 60.039 (sec). Leaf size: 43

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

\[ y(t)\to \int _1^t\left (e^{-3 K[2]} c_1+e^{-3 K[2]} \int _1^{K[2]}\delta (K[1])dK[1]\right )dK[2]+c_2 \]

✓ Sympy. Time used: 0.660 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{1} + \left (C_{2} - \frac {\int \operatorname {Dirac}{\left (t \right )} e^{3 t}\, dt}{3}\right ) e^{- 3 t} + \frac {\int \operatorname {Dirac}{\left (t \right )}\, dt}{3} \]