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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. In this tutorial, we are doing systems of two ODEs (hence <math>2×2</math> matrices involved) to reduce the work.
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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} </math>.
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 +
  In this tutorial, we are doing systems of two ODEs (hence <math>2×2</math> matrices involved) to reduce the work.
 
</font>
 
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'''<font size="4px"> 4.1 Homogeneous Linear Systems with Constant Coefficients </font>'''
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'''<font size="4px"> 4.1 ODE Systems with Real Eigenvalues </font>'''
  
 
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Revision as of 22:07, 18 November 2017

Systems of ODEs

A slecture by Yijia Wen

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} $.

 In this tutorial, we are doing systems of two ODEs (hence $ 2×2 $ matrices involved) to reduce the work.


4.1 ODE Systems with Real Eigenvalues


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.

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