Reduction formulae
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First order systems - Simultaneous differential equations - Systems of differential
A system is said to be first order if it contains no derivatives higher than the first order. It is linear if there are no combinations such as [Equation goes here - download the original to see it.] - so the functions of the dependant variables are added together and products of the dependant variables are not formed. If the entries in the matrices are constants then the system will be called constant coefficient. A linear constant-coefficient first-order system is said to be in normal form if it is written [Equation goes here - download the original to see it.] If the matrix [Equation goes here - download the original to see it.] then the system is said to be homogeneous. Every constant-coefficient linear first-order system of differential equations can be written in normal form, provided one condition is met. The general constant-coefficient linear first order system takes the form [Equation goes here - download the original to see it.] Then, is A is non-singular, that is, comprising of rows (or columns) of linear independent vectors- then A has to be capable of being put into normal form. So by multiplying all entries by [Equation goes here - download the original to see it.]. [Equation goes here - download the original to see it.] we get [Equation goes here - download the original to see it.] since [Equation goes here - download the original to see it.] Rearranging [Equation goes here - download the original to see it.] Then setting [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] we obtain [Equation goes here - download the original to see it.] which is in normal form.
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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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