70.28.3 problem 4
Internal
problem
ID
[19156]
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, October 02, 2025 at 03:38:46 PM
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: 0.235 (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 ) &= \cos \left (t \right ) c_1 +2 \cos \left (t \right ) t +\sin \left (t \right ) c_2 -\sin \left (t \right ) t -\cos \left (t \right ) \\
x_{2} \left (t \right ) &= \frac {2 \cos \left (t \right ) c_1}{5}-\frac {\cos \left (t \right ) c_2}{5}+\cos \left (t \right ) t +\frac {\sin \left (t \right ) c_1}{5}+\frac {2 \sin \left (t \right ) c_2}{5}-\cos \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.089 (sec). Leaf size: 146
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 -5 \sin (t) \int _1^t\left (2 \sin ^2(K[2])+\sin (2 K[2])\right )dK[2]+(2 \sin (t)+\cos (t)) \int _1^t(-3 \cos (2 K[1])+\sin (2 K[1])+2)dK[1]-5 c_2 \sin (t)+c_1 (2 \sin (t)+\cos (t))\\ \text {x2}(t)&\to (\cos (t)-2 \sin (t)) \int _1^t\left (2 \sin ^2(K[2])+\sin (2 K[2])\right )dK[2]+\sin (t) \int _1^t(-3 \cos (2 K[1])+\sin (2 K[1])+2)dK[1]+c_1 \sin (t)+c_2 (\cos (t)-2 \sin (t)) \end{align*}
✓ Sympy. Time used: 0.112 (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)
\[
\left [ x^{1}{\left (t \right )} = - t \sin ^{3}{\left (t \right )} + 2 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} - t \sin {\left (t \right )} \cos ^{2}{\left (t \right )} + 2 t \cos ^{3}{\left (t \right )} - \left (C_{1} - 2 C_{2}\right ) \cos {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \sin {\left (t \right )} - 3 \sin ^{3}{\left (t \right )} - 3 \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ x^{2}{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + t \cos ^{3}{\left (t \right )} - \sin ^{3}{\left (t \right )} - \sin {\left (t \right )} \cos ^{2}{\left (t \right )}\right ]
\]