76.28.3 problem 4
Internal
problem
ID
[17791]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
6.
Systems
of
First
Order
Linear
Equations.
Section
6.6
(Nonhomogeneous
Linear
Systems).
Problems
at
page
436
Problem
number
:
4
Date
solved
:
Thursday, March 13, 2025 at 10:49:21 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )-\cos \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\sin \left (t \right ) \end{align*}
✓ Maple. Time used: 0.374 (sec). Leaf size: 59
ode:=[diff(x__1(t),t) = 2*x__1(t)-5*x__2(t)-cos(t), diff(x__2(t),t) = x__1(t)-2*x__2(t)+sin(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_{2} \sin \left (t \right )-t \sin \left (t \right )+\cos \left (t \right ) c_{1} +2 t \cos \left (t \right )-\cos \left (t \right ) \\
x_{2} \left (t \right ) &= \frac {c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \sin \left (t \right )}{5}+\frac {2 \cos \left (t \right ) c_{1}}{5}-\frac {c_{2} \cos \left (t \right )}{5}+t \cos \left (t \right )-\cos \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.042 (sec). Leaf size: 61
ode={D[x1[t],t]==2*x1[t]-5*x2[t]-Cos[t],D[x2[t],t]==1*x1[t]-2*x2[t]+Sin[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \left (2 t-\frac {1}{2}+c_1\right ) \cos (t)-(t-1-2 c_1+5 c_2) \sin (t) \\
\text {x2}(t)\to (t-1+c_2) \cos (t)+\frac {1}{2} (1+2 c_1-4 c_2) \sin (t) \\
\end{align*}
✓ Sympy. Time used: 0.197 (sec). Leaf size: 116
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-2*x__1(t) + 5*x__2(t) + cos(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 2*x__2(t) - sin(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - t \sin ^{3}{\left (t \right )} + 2 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} - t \sin {\left (t \right )} \cos ^{2}{\left (t \right )} + 2 t \cos ^{3}{\left (t \right )} - \left (C_{1} - 2 C_{2}\right ) \cos {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \sin {\left (t \right )} - 3 \sin ^{3}{\left (t \right )} - 3 \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ x^{2}{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + t \cos ^{3}{\left (t \right )} - \sin ^{3}{\left (t \right )} - \sin {\left (t \right )} \cos ^{2}{\left (t \right )}\right ]
\]