First Order Systems of Linear Equations. or ODEs of Arbitrary Order

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1 First Order Systems of Linear Equations or ODEs of Arbitrary Order

2 Systems of Equations Relate Quantities Examples Predator-Prey Relationships r 0 = r (100 f) f 0 = f (r 50) (Lokta-Volterra Model)

3 Systems of Equations Relate Quantities Examples Physical Systems p 0 = v v 0 = k (p p 0 ) (Harmonic Oscillator) (Mass Spring System)

4 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 )

5 Linear vs Non-Linear First Order Linear Can Be Written As y 1 0 = a 11 (t)y 1 + a 12 (t)y a 1n (t)y n y 2 0 = a 21 (t)y 1 + a 22 (t)y a 2n (t)y n +g 1 (t) +g 2 (t)... y 0 n = a n1 (t)y 1 + a n2 (t)y a nn (t)y n +g n (t) Non-Linear Cannot Be Written This Way! If g(t) = 0 for all N, is homogeneous. Otherwise, non-homogeneous.

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

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

8 All Ordinary Differential Equations Are Systems of First Order Equations Consider The Second Order Equation y =0 w 0 y 0

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

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

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

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

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

14 All Ordinary Differential Equations Are Systems of First Order Equations The Second Order Equation 3y y 0 + 2y =0 Can Be Written As y 0 w 0 = = w 2 3 y 7 w 3

15 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, y 0, y 00, y 000,..., y (n 1)

16 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, y 00, y 000,..., y (n 1)

17 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, y 00, y 000,..., y v = w 0 (n 1)

18 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, y 00, y 000,..., y v = w 0 w 0 = y 00 (n 1)

19 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, y 00, y 000,..., y v = w 0 w 0 = y 00 (n 1)

20 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, y 00, y 000,..., y v = w 0 v = y 00 (n 1)

21 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, v, y 000,..., y v = w 0 v = y 00 (n 1)

22 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, v, y 000,..., y v = w 0 v = y 00 q = v 0 (n 1)

23 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, v, y 000,..., y (n 1) v = w 0 v = y 00 q = v 0 v 0 = y 000

24 We can do this in general Consider a General (Linear or Non-Linear) Equation of any Order y (n) = F t, y, w, v, y 000,..., y v = w 0 v = y 00 (n 1) q = v 0 q = y 000

25 We can do this in general Consider a General (Linear or Non-Linear) Equation y (n) = of any Order F t, y, w, v, q,..., v = w 0 v = y 00 q = v 0 q = y z = u 0 z

26 We can do this in general Consider a General (Linear or Non-Linear) Equation y (n) = of any Order F t, y, w, v, q,..., v = w 0 v = y 00 q = v 0 q = y z = u 0 z

27 We can do this in general Consider a General (Linear or Non-Linear) Equation y (n) = of any Order F t, y, w, v, q,..., v = w 0 v = y 00 q = v 0 q = y z = u 0 z 0 = y (n) z

28 We can do this in general Consider a General (Linear or Non-Linear) Equation z 0 of any Order = F t, y, w, v, q,..., v = w 0 v = y 00 q = v 0 q = y z = u 0 z 0 = y (n) z

29 We can do this in general Consider a General (Linear or Non-Linear) Equation z 0 y 0 of any Order = F t, y, w, v, q,..., z = w w 0 = v v 0 = q... u 0 = z

30 We can do this in general Consider a General (Linear or Non-Linear) Equation z 0 of any Order = F t, y, w, v, q,..., z Only first derivatives! y 0 = w w 0 = v v 0 = q... u 0 = z

31 We can do this in general Consider a General (Linear or Non-Linear) Equation z 0 of any Order = F t, y, w, v, q,..., z Only first derivatives! y 0 = w w 0 = v v 0 = q u 0... = z First Order (Linear or Non-Linear) System of Equations

32 Punchline Any Ordinary Differential Equation Of ANY Degree Is Really Just A System of First Order Ordinary Differential Equations!

33 So What? Existence: Solutions exist as long as F is continuous and has continuous derivatives Uniqueness: Only One Solution Exists

34 Summary All Differential Equation of Any Order Can Be Written As A System of First Order Differential Equations Systems Are Either Linear or Non-Linear We know unique solutions exist for any first order system as long as F has continuous derivatives.

35 Questions?