14.19.3 problem 6
Internal
problem
ID
[2696]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.12,
Dirac
delta
function.
Excercises
page
250
Problem
number
:
6
Date
solved
:
Sunday, March 30, 2025 at 12:15:00 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+y&=2 \delta \left (t -1\right )-\delta \left (t -2\right ) \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}
✓ Maple. Time used: 0.405 (sec). Leaf size: 71
ode:=diff(diff(y(t),t),t)+diff(y(t),t)+y(t) = 2*Dirac(t-1)-Dirac(t-2);
ic:=y(0) = 1, D(y)(0) = 0;
dsolve([ode,ic],y(t),method='laplace');
\[
y = -\frac {2 \left (\operatorname {Heaviside}\left (t -2\right ) \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (t -2\right )}{2}\right ) {\mathrm e}-2 \sqrt {3}\, {\mathrm e}^{\frac {1}{2}} \operatorname {Heaviside}\left (t -1\right ) \sin \left (\frac {\sqrt {3}\, \left (t -1\right )}{2}\right )-\frac {\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {3 \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}\right ) {\mathrm e}^{-\frac {t}{2}}}{3}
\]
✓ Mathematica. Time used: 0.233 (sec). Leaf size: 98
ode=D[y[t],{t,2}]+D[y[t],t]+y[t]==2*DiracDelta[t-1]-DiracDelta[t-2];
ic={y[0]==1,Derivative[1][y][0] ==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{3} e^{-t/2} \left (-2 \sqrt {3} e \theta (t-2) \sin \left (\frac {1}{2} \sqrt {3} (t-2)\right )+4 \sqrt {3 e} \theta (t-1) \sin \left (\frac {1}{2} \sqrt {3} (t-1)\right )+\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+3 \cos \left (\frac {\sqrt {3} t}{2}\right )\right )
\]
✓ Sympy. Time used: 5.630 (sec). Leaf size: 236
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(Dirac(t - 2) - 2*Dirac(t - 1) + y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0)
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\left (- \frac {2 \sqrt {3} \int \left (- \operatorname {Dirac}{\left (t - 2 \right )} + 2 \operatorname {Dirac}{\left (t - 1 \right )}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} - \frac {2 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} + \frac {4 \sqrt {3} \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} + 1\right ) \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {2 \sqrt {3} \int \left (- \operatorname {Dirac}{\left (t - 2 \right )} + 2 \operatorname {Dirac}{\left (t - 1 \right )}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} - \frac {2 \sqrt {3} \int \limits ^{0} \left (- \operatorname {Dirac}{\left (t - 2 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right )\, dt}{3} - \frac {2 \sqrt {3} \int \limits ^{0} 2 \operatorname {Dirac}{\left (t - 1 \right )} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\, dt}{3} + \frac {\sqrt {3}}{3}\right ) \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}}
\]