Systems of ODEs
4.0 Concept
Similar as systems of normal equations, several ODEs can also form a system. A typical system of $ n $
coupled first-order ODE looks like:
$ \frac{dx_1}{dt}=f_1(t,x_1,x_2,...x_n) $
$ \frac{dx_2}{dt}=f_2(t,x_1,x_2,...x_n) $
...
$ \frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n) $
To solve them, we introduce a method with eigenvectors and eigenvalues of matrices. There is an essential theorem for it. If $ \frac{dx}{dt}=A\bold{x} $, and the $ n×n $ matrix $ A $ has $ n $ distinct real eigenvalues with corresponding eigenvectors, the general solution will be $ \bold{x}=C_1 e^{\lambda_1 t} \bold{v_1}+C_2 e^{\lambda_2 t} \bold{v_2}+...+C_n e^{\lambda_n t} \bold{v_n} $, where $ \lambda_n $ are eigenvalues, $ \bold{v_n} $ are eigenvectors, and $ C_n $ are arbitrary constants. Strictly, the theorem is derived from the matrix exponential of the power series for $ e^A $, while we don't prove it here, but use a more intuitive explanation of analogy instead.
In one-dimensional space, a single linear ODE $ \frac{dx}{dt}=\lambda x $ has a solution $ x=Ae^{\lambda t} $, where $ \lambda $ is a constant. Similarly, in two(or more)-dimensional space, a linear ODE system $ \frac{dx}{dt}=A\bold{x} $ will have a solution in the form $ \bold{x}=e^{\lambda t} \bold{v} $, where $ A $ is a matrix without unknowns, $ \lambda $ is a constant and $ \bold{v} $ is a constant vector. For a matrix, the most identical constant and constant vector are going to be its eigenvalues and eigenvectors. In this tutorial, we are doing systems of two ODEs (hence $ 2×2 $ matrices involved) for examples.
First of all, we should be familiar with how to convert a system of linear equations to the matrix form. The same idea is used to convert a system of linear ODEs to the matrix form. For example, consider the system of linear ODEs
$ \frac{dx}{dt}=8x+2y $,
$ \frac{dy}{dt}=2x+5y $.
We separate the variables and their coefficients to get the matrix form $ \begin{bmatrix} \frac{dx}{dt}\\ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 8 & 2\\ 2 & 5 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $
From here we can start our journey.
4.1 ODE Systems with Real Eigenvalues
When you are given a matrix, the first thing to do is to find its identities, which is something distinguished it from anything else. The most intrinsic property for a matrix are its eigenvalues and eigenvectors.
Consider the theorem and system from 4.0, $ \begin{bmatrix} \frac{dx}{dt}\\ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 8 & 2\\ 2 & 5 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $. We can easily calculate the eigenvalues $ \lambda_1=4 $, $ \lambda_2=9 $, and therefore eigenvectors $ \bold{v_1}=\begin{bmatrix} 1\\ -2 \end{bmatrix} $, $ \bold{v_2}=\begin{bmatrix} 2\\ 1 \end{bmatrix} $. Plug them in the standard form of general solution in 4.0, we have the general solution to this system of linear ODEs is $ \bold{x}=C_1 e^{4t} \begin{bmatrix} 1\\ -2 \end{bmatrix} + C_2 e^{9t} \begin{bmatrix} 2\\ 1 \end{bmatrix} $, where $ \bold{x}=\begin{bmatrix} x\\ y \end{bmatrix} $.
Refer to here for further explanation of the phase portrait, an understanding from the geometrical perspective.
Sometimes the eigenvalues will be repeated, refer to here for a solution to this, as I feel like I can't explain more clear than it. :)
4.2 ODE Systems with Complex Eigenvalues
4.3 Exercises
4.4 References
Faculty of Mathematics, University of North Carolina at Chapel Hill. (2016). Linear Systems of Differential Equations. Chapel Hill, NC., USA.
Institute of Natural and Mathematical Science, Massey University. (2017). 160.204 Differential Equations I: Course materials. Auckland, New Zealand.
Robinson, J. C. (2003). An introduction to ordinary differential equations. New York, NY., USA: Cambridge University Press.
