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<math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n)</math> | <math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,...x_n)</math> | ||
| − | To solve them, we introduce a method with eigenvectors and eigenvalues of matrices. There is an essential theorem for it. '''If <math>\frac{dx}{dt}=A\bold{x}</math>, and the <math>n×n</math> matrix <math>A</math> has <math>n</math> distinct real eigenvalues with corresponding eigenvectors, the general solution will be <math>\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} </math>,''' where <math>\lambda_n</math> are eigenvalues, <math>\bold{v_n}</math> are eigenvectors, and <math>C_n</math> are arbitrary constants. | + | To solve them, we introduce a method with eigenvectors and eigenvalues of matrices. There is an essential theorem for it. '''If <math>\frac{dx}{dt}=A\bold{x}</math>, and the <math>n×n</math> matrix <math>A</math> has <math>n</math> distinct real eigenvalues with corresponding eigenvectors, the general solution will be <math>\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} </math>,''' where <math>\lambda_n</math> are eigenvalues, <math>\bold{v_n}</math> are eigenvectors, and <math>C_n</math> are arbitrary constants. In this tutorial, we are doing systems of two ODEs (hence <math>2×2</math> matrices involved) to reduce the work. |
| − | + | ||
| − | In this tutorial, we are doing systems of two ODEs (hence <math>2×2</math> matrices involved) to reduce the work. | + | |
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 | 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 | ||
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y \end{bmatrix} </math> | y \end{bmatrix} </math> | ||
| − | + | From here we can start our journey. :) </font> | |
| − | + | ||
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| − | </font> | + | |
Revision as of 00:08, 19 November 2017
Systems of ODEs
4.0 Abstract
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. In this tutorial, we are doing systems of two ODEs (hence $ 2×2 $ matrices involved) to reduce the work.
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
$ \begin{bmatrix} \frac{dx}{dt}\\ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 2 & 0\\ 0 & 3 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} $
4.2 Inhomogeneous Linear Systems
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.
