In Differential calculusDifferential calculus equation there were only differential equations in one dimension to be solved. But there are various occasions were a differential equation of more than just one dimension must be solved.
The equation
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where y and its differentiation appears in form of a vector and A is a matrix, is called a first order homogeneous differential equation system.
To solve this equation let
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With x as a vector of fixed values and λ a parameter.
Then
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And with the formulation from above:
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And y’ replaced
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The vector of y on the left side can be replaced as well:
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and with some brackets:
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And this is the basic equation for one Eigenvalue and Eigenvector of the matrix A (see Eigenvalues)
That means the equation
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Has one solution for the Eigenvalue λ and its Eigenvector x of the matrix A. And if the matrix A has more than one Eigenvalue, the differential equation system has more than one solution to. That means to solve this equation the Eigenvalues and their Eigenvectors of the matrix A must be found.
For a sample equation system like:
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The matrix form is
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And for the Eigenvalues the following Determinant shall be 0
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(See Eigenvalue calculation by the roots of the characteristic polynomial)
(o.k. that’s a very simple sample
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This Determinant is
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And it becomes 0 if λ = 1 or λ = 2 or λ = 3
To get the Eigenvectors these solutions must be inserted in the equation
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For λ = 1 that is
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These equations are under determined and so x1 = 1 can be set arbitrary.
And with this
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For λ = 2 that is
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These equations are under determined to. But x1 = 0 and x2 = 0. Only x3 can be set arbitrary. So we can set and
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For λ = 3 that is
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These equations are under determined again and so x2 = 1 can be set and
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So we have 3 solutions:
Whit λ = 1:
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Whit λ = 2:
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Whit λ = 3:
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To evaluate this result:
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For λ = 1:
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o.k.
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Such a simple equation can be solved manually. That’s nice to show. But for more complex equations a computer supported approach is required and therefore see Eigenvalue calculation by the roots of the characteristic polynomial)
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