# Lecture 12 - First Order Systems of Linear Equations or...

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First Order Systems of Linear Equations or ODEs of Arbitrary Order

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Systems of Equations Relate Quantities Examples Predator-Prey Relationships r 0 = r ( 100 - f ) f 0 = f ( r - 50 ) (Lokta-Volterra Model)
Systems of Equations Relate Quantities Examples Physical Systems p 0 = v v 0 = - k ( p - p 0 ) (Harmonic Oscillator) (Mass Spring System)

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Systems of Ordinary Differential Equations? A System of (First Order) Differential Equations: y 1 0 = F 1 ( y 1 , y 2 , y 3 , ... y n ) y 2 0 = F 2 ( y 1 , y 2 , y 3 , ... y n ) y 3 0 = F 3 ( y 1 , y 2 , y 3 , ... y n ) . . . y n 0 = F n ( y 1 , y 2 , y 3 , ..., y n )
Linear vs Non-Linear First Order Linear Can Be Written As y 1 0 = a 11 ( t ) y 1 + a 12 ( t ) y 2 + ... + a 1 n ( t ) y n y 2 0 = a 21 ( t ) y 1 + a 22 ( t ) y 2 + ... + a 2 n ( t ) y n y n 0 = a n 1 ( t ) y 1 + a n 2 ( t ) y 2 + ... + a nn ( t ) y n . . . Non-Linear Cannot Be Written This Way! + g 1 ( t ) + g 2 ( t ) + g n ( t ) If g(t) = 0 for all N, is homogeneous. Otherwise, non-homogeneous.

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All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y 00 y 0 y 3 7 2 + + = 0 y 0 w = = ( y 0 ) 0 = y 00 w 0
All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y 0 y 3 7 2 + + = 0 y 0 w = w 0 = ( y 0 ) 0 = y 00 w 0

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All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y 0 y 3 7 2 + + = 0 y 0 w = w 0
All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y 3 7 2 + + = 0 y 0 w = w 0 w

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All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y 0 w = y 7 2 = - - w 0 3 w
All Ordinary Differential Equations Are Systems of First Order Equations y = - - w 0 Consider The Second Order Equation y 0 w = w 2 3 7 3

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All Ordinary Differential Equations Are Systems of First Order Equations y = - - w 0 Consider The Second Order Equation y 0 w = w 2 3 7 3
All Ordinary Differential Equations Are Systems of First Order Equations y = - - w 0 Consider The Second Order Equation y 0 w = w 2 3 7 3

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