Introduction to ODE's (0A)
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A triangle and its slope y = f x, f f h f h f ( +h) f ( ) +h h, f h http://en.wikipedia.org/wiki/derivative, f Intro to ODEs (0A) 3
Many smaller triangles and their slopes f ( + h) f ( ) h ( +h, f ( +h)) f ( + h 1 ) f ( ) h 1 f ( + h 2 ) f ( ) h 2 f ( +h) f ( +h 1 ) f ( +h 2 ) f ( ) (, f ( )) +h 2 +h 1 +h h 2 h 1 lim h 0 f ( + h) f ( ) h h ( +h, f ( +h)) (, f ( )) http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 4
The limit of triangles and their slopes y = f x The derivative of the function f at The derivative function of the function f f ' ( ) = lim h 0 f ( + h) f ( ) h f '(x) = lim h 0 f (x + h) f (x) h y ' = f ' (x) = df dx = d dx f (x) http://en.wiktionary.org/ Intro to ODEs (0A) 5
The derivative as a function f x y = f x x 3 f f f x 3 Derivative Function y ' = f ' (x) f ' x f ' = lim h 0 f (x + h) f (x) h x 3 f ' f ' x 3 Intro to ODEs (0A) 6
The notations of derivative functions Largrange's Notation y ' = f ' x f ' (x) f ' Leibniz's Notation f ' dy dx = d dx f (x) x 3 f ' x 3 not a ratio. Newton's Notation ẏ = ḟ x slope of a tangent line lim h 0 f (x + h) f (x) h Euler's Notation D x y = D x f x derivative with respect to x x is an independent variable Intro to ODEs (0A) 7
Another kind of triangles and their slopes y = f x y ' = f ' x given x1 f the slope of a tangent line http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 8
Differential in calculus Differential: dx, dy, infinitesimals a change in the linearization of a function of, or relating to differentiation function f dx the linearization of a function f dy f dx dx dx f http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 9
Approximation Differential: dx, dy, f ( + dx) f ( ) + dy = f ( ) + f '( )dx f '( ) = lim h 0 f ( + h) f ( ) h function the linearization of a function f dx f dy f dx dx dx f http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 10
Differential as a function Line equation in the new coordinate. dy dy = f ' ( ) dx slope = f ' ( ) f ( ) dx http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 11
Differential as a function The differential of a function ƒ(x) of a single real variable x is the function of two independent real variables x and dx given by dy = f ' (x) dx Line equation in the new coordinate. dy dy = f ' dx slope = f ' (x, dx) dy f dx http://en.wikipedia.org/wiki/derivative Intro to ODEs (0A) 12
Differentials and Derivatives (1) differentials ratio dy = f ' (x) dx dy = df dx dx derivative dy dx = f ' (x) not a ratio f ( + dx) f ( ) + dy for small enough dx = f ( ) + f '( )dx f ( + dx) = f ( ) + dy f ( + dx) f ( ) dx = f ( ) + f '( )dx = f '( ) Intro to ODEs (0A) 13
Differentials and Derivatives (2) dy = f ' (x) dx dy = df dx dx dy = ḟ dx dy = f '(x) dx dy = df dx dx dy = 1dy = y dy = D x f dx Intro to ODEs (0A) 14
Differentials and Derivatives (3) dy = f '(x) dx dy = f '(x) dx + C dy = df dx dx dy = df dx dx + C a constant is included another constant is included differ by a constant y + C 1 = f (x) + C 2 y = f (x) + C y = f (x) + C Intro to ODEs (0A) 15
Applications of Differentials (1) Substitution Rule f ( g(x) ) g'(x) dx = f ( u ) du (I) u = g( x) du = g ' (x)dx du = dg dx dx (II) f (g) dg dx dx = f ( g ) dg Intro to ODEs (0A) 16
Applications of Differentials (2) Integration by parts f (x)g ' (x) dx = f (x)g(x) f ' (x)g(x) dx u = f (x) v = g(x) du = f ' (x) dx dv = g'( x)dx du = df dx dx dv = dg dx dx f (x)g ' (x) dx = f (x)g(x) f ' (x)g(x) dx u dv = u v v du Intro to ODEs (0A) 17
Anti-derivative and Indefinite Integral F '(x) = f (x) F(x) Anti-derivative without constant the most simple anti-derivative F( x) + C the most general anti-derivative f (x)dx Indefinite Integral : a function of x f (x)dx = F (x) + C Intro to ODEs (0A) 18
Anti-derivative Examples (1)? Anti-derivative of f(x) differentiation Anti-differentiation derivative of f (x)? F 1 (x)= 1 3 x3 f (x)= All are Anti-derivative of f(x) F 2 (x)= 1 3 x3 + 100 F 3 (x)= 1 3 x3 49 the most general anti-derivative of f(x) 1 3 x3 + C dx indefinite Integral of f(x) Intro to ODEs (0A) 19
Anti-derivative Examples (2)? Anti-derivative of f(x) differentiation Anti-differentiation derivative of? f (x) x 0 f (x) dx = [ 1 3 x3]0 x = 1 3 x3 f (x)= x a f (x) dx = [ 1 3 x3]a x = 1 3 x3 1 3 a3 x a f (t ) dt = [ 1 3 t3]a x = 1 3 x3 1 3 a3 anti-derivative by the definite integral of f(x) x a t 2 dt = 1 3 x3 + C d x dx a f (t ) dt = f (x) = the Indefinite Integral of f(x) dx = 1 3 x3 + C Intro to ODEs (0A) 20
Definite Integrals on [a, ] 1 y = 1 1 a a a f ' (x) = 1 f (x) = 1 a 1 dx a 1 dx [ x ] a = a dx dy = dy dx dx view (I) view (II) a a f ' (x) dx [ f (x) ] a = f ( ) f (a) f (x) dx [ F (x)] a = F ( ) F (a) Intro to ODEs (0A) 21
Definite Integrals on [, ] view (I) view (II) f (x) F (x) : anti-derivative of f '(x) : anti-derivative of f ( x) 1 dx Anti-derivative x 1 y = 1 area Anti-derivative without constant a [ x + c ] x1 = = [ x ] x1 length view (I) [ f (x) ] x1 = f ( ) f ( ) x view (II) [ F (x)] x1 = F ( ) F ( ) a Intro to ODEs (0A) 22
Definite Integrals on [, ] through F(x)+C view (I) Anti-derivative view (II) Anti-derivative 1 dx f (x) = x 1 dx F (x) = x f '(x) f (x) = [ f (x)] x1 = [ F (x)] x1 = {f ( ) f (a)} {f ( ) f (a)} = {F ( ) F(a)} {F ( ) F(a)} = [ f (x) f (a)] x1 = [ F (x) F (a) ] x1 = [ f (x) + C ] x1 arbitrary reference point (a, f(a)) = [ F (x) + C ] x1 arbitrary reference point (a, F(a)) = [ f ' (x)dx ] x1 = [ f (x)dx ] x1 Intro to ODEs (0A) 23
Indefinite Integrals through Definite Integrals view (I) view (II) 1 dx x a 1 dx 1 dx x a 1 dx f ' (x) dx a f ' (x) dx f (x) dx a f (x) dx = f (x) f (a) = x a = f (x) + C = F(x) + C = F(x) F (a) = x a a a f (x) = x a F (x) = x a a a a a Intro to ODEs (0A) 24
Definite Integrals on [a, ] and [a, ] (I) a f '(x) dx (II) a f (x) dx a (I) a f '(x) dx (II) a f (x) dx a Intro to ODEs (0A) 25
Definite Integrals on [, ] through F(x) d f d x dx Anti-derivative F (x) = f (x) y = f ' (x) area a f ( ) f ( ) = [ F (x)] x1 F (x) = f (x) F ( ) F ( ) length a Intro to ODEs (0A) 26
Definite Integrals on [, ] through F(x)+C d f d x dx F (x) = f (x) y = f ' (x) a common reference point a f ( ) f ( ) = [ F (x)] x1 = {F ( ) F(a)} {F ( ) F(a)} = [ F (x) F (a)] x1 = [ F (x) C ] x1 = [ f (x) f (a)] x1 arbitrary reference point (a, F(a)) Intro to ODEs (0A) 27
Indefinite Integrals through F(x)+C d f d x dx = F ( x) + C x a d f d x dx y = f ' (x) a common reference point a F (x) F (a) = F (x) f (a) f ( ) f (a) y = f (x) f (a) a Intro to ODEs (0A) 28
Indefinite Integrals a 1 dx x a 1 dx dx dy a x a x + C y + C given a variable x indefinite integral a d f d x dx x c d f d x dx d f d x dx f ( ) f (a) f (x) f (a) f (x) + C given a variable x indefinite integral Intro to ODEs (0A) 29
Derivative Function and Indefinite Integrals f ' ( ) lim h 0 f ( + h) f ( ) h f ( x) dx f ' ( ) lim h 0 f ( + h) f ( ) h x 4 x 3 f ( x) dx f ' (x 3 ) lim h 0 f (x 3 + h) f (x 3 ) h x 6 x 5 f ( x) dx,, x 3 [, ],[ x 3, x 4 ], [x 5, x 6 ] f ' (x) = lim h 0 f (x + h) f ( x) h x F (x) + C = a f ( x) dx f ' ( ), f ' ( ), f '( x 3 ) [ F( x) ], [ F( x) ]x 3 x 4 x, [ F (x) 6 ]x5 Intro to ODEs (0A) 30
a Intro to ODEs (0A) 31
Differential Equation f (x) = e 3 x f (x)? f ' (x) = 3e 3 x f ' (x) 3 f (x) = 3 e 3x 3 e 3 x = 0 f ' (x) 3 f (x) = 0 Intro to ODEs (0A) 32
First Order Examples (y=f(x)) f ' (x) = f (x) y ' = y f (x)? y? An Example of A First Order Differential Equation f (x) = c e x for all x y = c e x I : (, + ) f ' (x) = f (x) y ' = y f (x)? y? f (0) = 3 f (0) = 3 An Example of A First Order Initial Value Problem f (x) = 3e x for all x y = 3e x I : (, + ) Intro to ODEs (0A) 33
Second Order Examples (y=f(x)) f ' ' (x) = f (x) y ' ' = y f (x) = y = An Example of A Second Order Differential Equation c 1 e +x + c 2 e x for all x c 1 e +x + c 2 e x I : (, + ) f ' ' (x) = f (x) y ' ' = y f (x) = y = f ' (0) = 0 y ' (0) = 0 f (0) = 1 f (0) = 1 +1 e +x 1 e x +1 e +x 1 e x An Example of A Second Order Initial Value Problem for all x I : (, + ) Guess the possible solution. Intro to ODEs (0A) 34
General First & Second Order IVPs (y=f(x)) First Order Initial Value Problem f ' (x) = f (x) y ' = y f (0) = 3 f (0) = 3 d y = g(x, y) y ' = g(x, y) d x y(x 0 ) = y 0 y(x 0 ) = y 0 Second Order Initial Value Problem f ' ' (x) = f (x) y ' ' = y f ' (0) = 0 y ' (0) = 0 f (0) = 1 f (0) = 1 d 2 y d = g(x, y, y ') y(x 0 ) = y 0 y ' (x 0 ) = y 1 y ' ' = g(x, y, y ' ) y(x 0 ) = y 0 y ' (x 0 ) = y 1 Guess the possible solution. Intro to ODEs (0A) 35
References [1] http://en.wikipedia.org/ [2] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics