8.26.9 problem 9
Internal
problem
ID
[2782]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
3.
Systems
of
differential
equations.
Section
3.13
(Solving
systems
by
Laplace
transform).
Page
370
Problem
number
:
9
Date
solved
:
Tuesday, September 30, 2025 at 05:52:38 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+\delta \left (t -\pi \right ) \end{align*}
With initial conditions
\begin{align*}
x_{1} \left (0\right )&=1 \\
x_{2} \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.213 (sec). Leaf size: 58
ode:=[diff(x__1(t),t) = 2*x__1(t)-2*x__2(t), diff(x__2(t),t) = 4*x__1(t)-2*x__2(t)+Dirac(t-Pi)];
ic:=[x__1(0) = 1, x__2(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
x_{1} \left (t \right ) &= \cos \left (2 t \right )+\sin \left (2 t \right )-\sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \\
x_{2} \left (t \right ) &= \operatorname {Heaviside}\left (t -\pi \right ) \cos \left (2 t \right )-\sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \sin \left (2 t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 54
ode={D[x1[t],t]==2*x1[t]-2*x2[t]+0,D[x2[t],t]==4*x1[t]-2*x2[t]+DiracDelta[t-Pi]};
ic={x1[0]==1,x2[0]==0};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to -\theta (t-\pi ) \sin (2 t)+\sin (2 t)+\cos (2 t)\\ \text {x2}(t)&\to \theta (t-\pi ) (\cos (2 t)-\sin (2 t))+2 \sin (2 t) \end{align*}
✓ Sympy. Time used: 0.872 (sec). Leaf size: 211
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-2*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-Dirac(t - pi) - 4*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) \cos {\left (2 t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) \sin {\left (2 t \right )} - \frac {\sin {\left (2 t \right )} \int \left (- \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt}{2} - \frac {\sin {\left (2 t \right )} \int \left (\operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt}{2} - \frac {\cos {\left (2 t \right )} \int \left (- \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt}{2} + \frac {\cos {\left (2 t \right )} \int \left (\operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt}{2}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )} - \sin {\left (2 t \right )} \int \left (- \operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt + \cos {\left (2 t \right )} \int \left (\operatorname {Dirac}{\left (t - \pi \right )} \sin {\left (2 t \right )} + \operatorname {Dirac}{\left (t - \pi \right )} \cos {\left (2 t \right )}\right )\, dt\right ]
\]