Reduction formulae
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Example - Simultaneous differential equations - Systems of differential equations
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One of these two systems can be place into normal form. Determine which of these two systems can be placed into normal form; explain why the other cannot and find the normal form of the one that can have a normal form. (i) [Equation goes here - download the original to see it.] (ii) [Equation goes here - download the original to see it. Solution For system (i) the matrix form is [Equation goes here - download the original to see it.] We will see immediately in the matrix [Equation goes here - download the original to see it.] That the second row [Equation goes here - download the original to see it.] is a multiple of the first row [Equation goes here - download the original to see it. so A is non-singular and does not have an inverse. For system the matrix form is [Equation goes here - download the original to see it.] We need to find the inverse of [Equation goes here - download the original to see it.] We will find this inverse by the formula If [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] which we can check [Equation goes here - download the original to see it.] as expected. Our system is [Equation goes here - download the original to see it.] Where [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] Then premultiplying all the matrices by Simultaneous differential equations - Systems of differential equations [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Hence, in normal form the system is [Equation goes here - download the original to see it.]
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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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