8.26.10 problem 10
Internal
problem
ID
[2783]
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
:
10
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 )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )+1-\operatorname {Heaviside}\left (t -\pi \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}
With initial conditions
\begin{align*}
x_{1} \left (0\right )&=0 \\
x_{2} \left (0\right )&=0 \\
\end{align*}
✓ Maple. Time used: 0.129 (sec). Leaf size: 107
ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t)+1-Heaviside(t-Pi), diff(x__2(t),t) = 2*x__1(t)-2*x__2(t)];
ic:=[x__1(0) = 0, x__2(0) = 0];
dsolve([ode,op(ic)]);
\begin{align*}
x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-t}}{3}+\frac {2 \,{\mathrm e}^{2 t}}{3}+\operatorname {Heaviside}\left (t -\pi \right )-\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }}{3}-1-\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }}{3} \\
x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{3}+\frac {2 \,{\mathrm e}^{-t}}{3}+\operatorname {Heaviside}\left (t -\pi \right )-\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }}{3}-1-\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }}{3} \\
\end{align*}
✓ Mathematica. Time used: 11.254 (sec). Leaf size: 294
ode={D[x1[t],t]==3*x1[t]-2*x2[t]+1-HeavisideTheta[t-Pi],D[x2[t],t]==2*x1[t]-2*x2[t]};
ic={x1[0]==0,x2[0]==0};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to \frac {1}{9} e^{-3 t-2 \pi } \left (\left (4 e^{3 t}-1\right ) \left (-2 e^{2 t}-e^{2 t+3 \pi }+e^{3 t+2 \pi }+2 e^{2 \pi }\right ) \theta (t-1,t-\pi )+2 \left (e^{3 t}-1\right ) \left (e^{2 t}+2 e^{2 t+3 \pi }-2 e^{3 t+2 \pi }-e^{2 \pi }\right ) \theta (t-1) \theta (t-\pi )+3 e^{2 (t+\pi )} \left (-3 e^t+2 e^{3 t}+1\right )\right )\\ \text {x2}(t)&\to \frac {1}{9} e^{-3 t-2 \pi } \left (2 \left (e^{3 t}-1\right ) \left (-2 e^{2 t}-e^{2 t+3 \pi }+e^{3 t+2 \pi }+2 e^{2 \pi }\right ) \theta (t-1,t-\pi )+\left (e^{3 t}-4\right ) \left (e^{2 t}+2 e^{2 t+3 \pi }-2 e^{3 t+2 \pi }-e^{2 \pi }\right ) \theta (t-1) \theta (t-\pi )+3 e^{2 (t+\pi )} \left (e^t+2\right ) \left (e^t-1\right )^2\right ) \end{align*}
✓ Sympy. Time used: 0.457 (sec). Leaf size: 119
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-3*x__1(t) + 2*x__2(t) + Heaviside(t - pi) + Derivative(x__1(t), t) - 1,0),Eq(-2*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)
\[
\left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{2} + 2 C_{2} e^{2 t} - \frac {2 e^{2 t} \theta \left (t - \pi \right )}{3 e^{2 \pi }} + \theta \left (t - \pi \right ) - 1 - \frac {e^{\pi } e^{- t} \theta \left (t - \pi \right )}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{2 t} - \frac {e^{2 t} \theta \left (t - \pi \right )}{3 e^{2 \pi }} + \theta \left (t - \pi \right ) - 1 - \frac {2 e^{\pi } e^{- t} \theta \left (t - \pi \right )}{3}\right ]
\]