Introduction to ODE's (0P) Young Won Lim 12/27/14

Transcription

1 Introuction to ODE's (0P)

2 Copyright (c) Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any later version publishe by the Free Software Founation; with no Invariant Sections, no Front-Cover Texts, an no Back-Cover Texts. A copy of the license is inclue in the section entitle "GNU Free Documentation License". Please sen corrections (or suggestions) to youngwlim@hotmail.com. This ocument was prouce by using OpenOffice an Octave.

3 Derivative an Integral of Trigonometric Functions Intro to ODEs (0P) 3

4 Differentiation & Integration of sinusoial functions x f x = cos x leas f x = sin x x f x = sin x leas f x = cos x f x x = cos x C lags f x = sin x f x x = sin x C lags f x = cos x Intro to ODEs (0P) 4

5 Derivative of sin(x) f (x) = sin(x) slope leas x f (x) = cos(x) Intro to ODEs (0P) 5

6 Plot of F(x,y)=cos(x) f (x) = sin( x) slope F (x, y) slope=+1 F (x, f (x )) = f '( x) (x i, y i ) f '(x i ) f ' (x) = cos( x) Intro to ODEs (0P) 6

7 Plot of F(x,y)=cos(x) (x, y) = (x, f ( x)) = ( x,sin( x)) x y f (x) = sin( x) slope (x, y) f (x) = sin( x) slope m (x, y) m = slope of a tangent f ' (x) F (x, y) = f ' ( x) F (x, sin(x)) = cos(x) Intro to ODEs (0P) 7

8 Derivative of cos(x) f x = cos x slope leas x f x = sin x Intro to ODEs (0P) 8

9 Plot of F(x,y)=-sin(x) f (x ) = cos( x) slope F (x, y) slope=+1 F (x, f (x )) = f '( x) (x i, y i ) f '(x i ) f ' (x) = sin( x) Intro to ODEs (0P) 9

10 Integral of sin(x) f x = sin x 0 / 2 sin t t = 1 C = 1 0 x sin (t) t area + 0 C = area - 1 f x x = cos x C lags 0 x sin (t) t 1 = cos x Intro to ODEs (0P) 10

11 Integral of cos(x) f x = cos x 0 / 2 cos x x = area lags 0 x cos t t = + sin(x) f x x = sin x C Intro to ODEs (0P) 11

12 Derivative an Integral of Exponential Functions Intro to ODEs (0P) 12

13 The Euler constant e x ax = lim h 0 a x h a x h = a x lim h 0 a h 1 h a x such a, we call e lim h 0 a h 1 h a h a 0 = 1 lim f ' 0 = 1 h 0 h 0 = 1 x ex = e x e = f x = e x f ' x = e x f ' ' x = e x Intro to ODEs (0P) 13

14 The Euler constant e x ex = e x e = f x = e x f ' x = e x f ' ' x = e x lim h 0 a h 1 h f ' 0 = 1 = 1 iif a = e lim h 0 a h a 0 h 0 = 1 Functions f(x) = a x are shown for several values of a. e is the unique value of a, such that the erivative of f(x) = a x at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2 x (otte curve) an 4 x (ashe curve) are shown; they are not tangent to the line of slope 1 an y-intercept 1 (re). Intro to ODEs (0P) 14

15 The Derivative of a x a x = e ln ax = e x ln a x {ax } = x {e x ln a } = {e x ln a } x x {ax } = {a x } ln a {x ln a} x {ex } = {e x } ln e = {e x } Intro to ODEs (0P) 15

16 Differentiation an Integration (1) f (x) f ' (x)x = f ' (x) + c 1 x x f ' (x) f ' (x) e x e x + c x x e x e x Intro to ODEs (0P) 16

17 Differentiation an Integration (2) x f (x) f ' (x) f ' ' (x) x f '(x)x x f ' ' (x)x x f (3) (x)x = f (x) + c 1 = f '(x) + c 2 = f ' ' (x) + c 3 x x e x e x e x e x + c 1 x + c 2 x e x + c 1 x e x Intro to ODEs (0P) 17

18 Differentiation an Integration (3) f (x) f (x) x x f ' (x) F (x) + c x x f (x) + c f (x) e x e x x x e x e x + c x x e x + c e x Intro to ODEs (0P) 18

19 Chain Rule Intro to ODEs (0P) 19

20 Chain Rule f (g(x)) x f ' (g(x)) g' (x) f x = f g g x f g = f ' (g( x)) g x = g ' (x) f (g(x)) x f ' (g(x)) g' (x) with respect to with respect to x e P(x )x x e P(x )x x ( P(x)x ) = e P(x)x P(x) e g x e g g x f g g x Intro to ODEs (0P) 20

21 Substitution Rule Intro to ODEs (0P) 21

22 Substitution Rule f (g(x)) + C x f ' (g(x)) g' (x) f (g(x)) + C = f ' (g(x)) g'(x)x f (u) + C = f ' (u) u f '(g(x)) g ' (x) x = f '(g (x)) g x x = f '(u)u = f (u) + C u=g(x) u = g x x Intro to ODEs (0P) 22

23 Chain Rule an Substitution Rule Examples Intro to ODEs (0P) 23

24 Chain Rule an Substitution Rule f (g(x)) f x x = f ' (g(x)) g' (x) f g g x f (g(x)) + C x f ' (g(x)) g' (x) f (g(x)) + C = f ' (g(x)) g'(x)x Intro to ODEs (0P) 24

25 Substitution Rule Examples (1) f (g(x)) + C x f ' (g(x)) g' (x) f (g(x)) + C = f ' (g(x)) g'(x)x Ex 1: Ex 2: e 3 x x e g( x) g( x) g'(x) x = e g(x) = 3 x g'(x) = 3 e 2 y y e h ( y) h( y) h ' ( y) y = e h( y) = 2 y h'( y) = 2 e 3 x x = 1 3 e 3 x 3 x e 3 x x = 1 3 e3 x e 2 y y = 1 2 e 2 y 2 y e 2 y y = 1 2 e2 y Intro to ODEs (0P) 25

26 Substitution Rule Examples (2) f (g(x)) + C x f ' (g(x)) g'(x) f (g(x)) + C = f ' (g(x)) g'(x)x Ex 3: x (x 2 9) x = x(x 2 9) 1 x Ex 4: p( y) y x = g(x) y = Φ(x) = 1 2 (x 2 9) 1 2 x x p(φ(x))φ' (x) = g(x) = 1 2 (x 2 9) 1 { x (x2 9) } x = 1 u 1 u = 1 ln u = ln u 1/2 2 2 = ln(x 2 9) 1/2 = ln x 2 9 p(φ(x))φ' (x)x = g(x)x p( y)y = g(x)x y = Φ' (x)x for (x 2 >9) Intro to ODEs (0P) 26

27 Substitution Rule Examples (3) f (g(x)) + C x f ' (g(x)) g'(x) f (g(x)) + C = f ' (g(x)) g'(x)x Ex 5: x (x 1) 2 x = (x 1)+1 (x 1) 2 = u+1 u 2 u { x (x 1) } x = 1 u + 1 u 2 u = ln u 1 u + C Intro to ODEs (0P) 27

28 Derivative Prouct an Quotient Rule Intro to ODEs (0P) 28

29 Derivative Prouct an Quotient Rule f g x x (f g) = f ' g + f g' f x g + f g x f (x), g(x) f g x f ' g f g' g 2 f (x), g(x) ( f ) = ( f x g x g f g ) x / g2 Intro to ODEs (0P) 29

30 Integration By Parts Intro to ODEs (0P) 30

31 Integration by parts (1) f (x)g(x) x f ' (x)g(x) + f ( x) g' (x) x (f g) = f x g + f g x f (x)g(x) x f ' (x)g(x) + f ( x) g' (x) f g = f ' g x + f g ' x f (x)g ' (x) x = f (x)g(x) f ' (x)g(x) x Intro to ODEs (0P) 31

32 Integration by parts (2) f (x)g ' (x) x = f (x)g(x) f ' (x)g(x) x x e x x = x e x e x x = x e x e x + c 1 = ( x 1)e x + c 1 x 2 e x x = x 2 e x 2 xe x x = x 2 e x 2 x e x + 2e x + c 2 = ( x 2 2 x + 2) e x + c 2 x 3 e x x = x 3 e x 3 x 2 e x x = x 3 e x 3 x 2 e x + 6 x e x 6 e x + c 3 = ( x 3 3 x x 6)e x + c 3 x e x x x 2 e x x x 3 e x x = ( x 1)e x + c 1 e x x = { x ex }x = e x + c = ( x 2 2 x + 2) e x x + c 2 x e 2 /2 x = { /2 }x = e x2 /2 + c x ex2 = ( x 3 3 x x 6)e x + c 3 x 2 e x 3 /3 x = { /3 }x = e x3 /3 + c x ex3 Intro to ODEs (0P) 32

33 Derivative of Inverse Functions Intro to ODEs (0P) 33

34 Derivatives of Inverse Functions b b a a a b a y = f (x) b y = f 1 (x) = g(x) b (a, b) = (a, f (a)) a (b, a) = (b, f 1 (b)) = (b,g(b)) a b m 1 = f ' (a) m 2 = g '(b) m 1 m 2 = f '(a)g'(b) = 1 g '(b) = 1 f '(a) g '(b) = 1 f '(g(b)) g(b) = a Intro to ODEs (0P) 34

35 Derivatives of Inverse Functions m 1 m 2 = f '(a)g'(b) = 1 g '(b) = 1 f '(a) g '(b) = 1 f '(g(b)) To fin g'(x) (1) fin f'(x) (2) fin 1 / f'(x) (3) substitute x with g(x) g(b) = a g'(x) = 1 f ' (g(x)) x ln x 1 x x ex e x (e x )' e x 1 (ln x)' = 1 x 1 1/ x = x e x 1 e ln x = 1 x e x Intro to ODEs (0P) 35

36 Derivative of ln x f (x) = e x y = ln x g(x) = ln x g '(x) = 1 f ' (g(x)) e y = x x ln x 1 x (e x )' e x x e y = x x e y y x = 1 1 y x = 1 e y e x 1 e ln x = 1 x = 1 x e y = x Intro to ODEs (0P) 36

37 Derivative of ln x x ln x = 1 x x ln ( x) = 1 ( x) ( 1) = 1 x x>0 x<0 ln x x>0 ln( x) x<0 x x 1 x 1 x x ln x = 1 x x 0 Intro to ODEs (0P) 37

38 Inefinite Integral of ln x x ln x = 1 x x 0 ln x ln( x) (x>0) (x<0) x 1 x ln x = 1 x x (x 0) Intro to ODEs (0P) 38

39 References [1] [2] M.L. Boas, Mathematical Methos in the Physical Sciences [3] E. Kreyszig, Avance Engineering Mathematics [4] D. G. Zill, W. S. Wright, Avance Engineering Mathematics