Reduction formulae
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Solution to first-order constant coefficient homogeneous systems
A first order constant coefficient system of differential equations has the form [Equation goes here - download the original to see it.] If the system is also homogeneous then [Equation goes here - download the original to see it.] ; hence the form is [Equation goes here - download the original to see it.] The solution to the system is given by the following theorem. If Q has n linearly independent eigenvectors, [Equation goes here - download the original to see it.] corresponding to eigenvalues Simultaneous differential equations - Systems of differential equations (where the eigenvalues need not be distinct) then the solution to [Equation goes here - download the original to see it.] is [Equation goes here - download the original to see it.] where Simultaneous differential equations - Systems of differential equations are constants. Before we prove this theorem we will illustrate its application. Firstly, note that the theorem requires us to begin by placing the system in the form [Equation goes here - download the original to see it.] We then proceed by finding the eigenvectors and eigenvalues of the matrix Q. We conclude by substituting into the equation [Equation goes here - download the original to see it.] Example: Solve the system [Equation goes here - download the original to see it. Solution: The matrix form is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] So the system is [Equation goes here - download the original to see it.] Which in normal form is [Equation goes here - download the original to see it.] or Simultaneous differential equations - Systems of differential equations where [Equation goes here - download the original to see it.] we now proceed to find the eigenvalues and eigenvectors of . The eigenvalues satisfy the equation. Simultaneous differential equations - Systems of differential equations That is [Equation goes here - download the original to see it.] For Simultaneous differential equations - Systems of differential equations let the eigenvector be [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] For Simultaneous differential equations - Systems of differential equations let the eigenvector be [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] In summary for [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Substituting into Simultaneous differential equations - Systems of differential equations we obtain [Equation goes here - download the original to see it. where Simultaneous differential equations - Systems of differential equations are constants. Uncoupling this gives [Equation goes here - download the original to see it.] We will check the solution by substituting back into the original equation which was [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Substituting into the left hand side of (1) [Equation goes here - download the original to see it.] Substituting into the LHS of (2) [Equation goes here - download the original to see it.] So the solution is correct. The theorem will also apply if their are repeated eigenvalues, provided there are sufficient distinct eigenvectors. That is, so long as the matrix Q in Simultaneous differential equations - Systems of differential equations is non-singular
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Contents of
Reduction formulae
1 Reduction formulae
2 Simultaneous differential equations - Systems of differential equations
3 Matrix notation- Simultaneous differential equations - Systems of differential equat
4 First order systems - Simultaneous differential equations - Systems of differential
5 Example - Simultaneous differential equations - Systems of differential equations
6 Solution to first-order constant coefficient homogeneous systems
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