Introduction to ODE's (0A)
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A tringle nd its slope y = f x, f f h f h f ( +h) f ( ) +h h, f h http://en.wikipedi.org/wiki/derivtive, f Intro to ODEs (0A) 3
Mny smller tringles nd 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.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 4
The limit of tringles nd their slopes y = f x The derivtive of the function f t The derivtive 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.wiktionry.org/ Intro to ODEs (0A) 5
The derivtive s function f x y = f x x 3 f f f x 3 Derivtive 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 nottions of derivtive functions Lrgrnge's Nottion y ' = f ' x f ' (x) f ' Leibniz's Nottion f ' dy dx = d dx f (x) x 3 f ' x 3 not rtio. Newton's Nottion ẏ = ḟ x slope of tngent line lim h 0 f (x + h) f (x) h Euler's Nottion D x y = D x f x derivtive with respect to x x is n independent vrible Intro to ODEs (0A) 7
Another kind of tringles nd their slopes y = f x y ' = f ' x given x1 f the slope of tngent line http://en.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 8
Differentil in clculus Differentil: dx, dy, infinitesimls chnge in the lineriztion of function of, or relting to differentition function f dx the lineriztion of function f dy f dx dx dx f http://en.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 9
Approximtion Differentil: dx, dy, f ( + dx) f ( ) + dy = f ( ) + f '( )dx f '( ) = lim h 0 f ( + h) f ( ) h function the lineriztion of function f dx f dy f dx dx dx f http://en.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 10
Differentil s function Line eqution in the new coordinte. dy dy = f ' ( ) dx slope = f ' ( ) f ( ) dx http://en.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 11
Differentil s function The differentil of function ƒ(x) of single rel vrible x is the function of two independent rel vribles x nd dx given by dy = f ' (x) dx Line eqution in the new coordinte. dy dy = f ' dx slope = f ' (x, dx) dy f dx http://en.wikipedi.org/wiki/derivtive Intro to ODEs (0A) 12
Differentils nd Derivtives (1) differentils rtio dy = f ' (x) dx dy = df dx dx derivtive dy dx = f ' (x) not rtio f ( + dx) f ( ) + dy for smll enough dx = f ( ) + f '( )dx f ( + dx) = f ( ) + dy f ( + dx) f ( ) dx = f ( ) + f '( )dx = f '( ) Intro to ODEs (0A) 13
Differentils nd Derivtives (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
Differentils nd Derivtives (3) dy = f '(x) dx dy = f '(x) dx + C dy = df dx dx dy = df dx dx + C constnt is included nother constnt is included differ by constnt y + C 1 = f (x) + C 2 y = f (x) + C y = f (x) + C Intro to ODEs (0A) 15
Applictions of Differentils (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
Applictions of Differentils (2) Integrtion by prts 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-derivtive nd Indefinite Integrl F '(x) = f (x) F(x) Anti-derivtive without constnt the most simple nti-derivtive F( x) + C the most generl nti-derivtive f (x)dx Indefinite Integrl : function of x f (x)dx = F (x) + C Intro to ODEs (0A) 18
Anti-derivtive Exmples (1)? Anti-derivtive of f(x) differentition Anti-differentition derivtive of f (x)? F 1 (x)= 1 3 x3 f (x)= All re Anti-derivtive of f(x) F 2 (x)= 1 3 x3 + 100 F 3 (x)= 1 3 x3 49 the most generl nti-derivtive of f(x) 1 3 x3 + C dx indefinite Integrl of f(x) Intro to ODEs (0A) 19
Anti-derivtive Exmples (2)? Anti-derivtive of f(x) differentition Anti-differentition derivtive of? f (x) x 0 f (x) dx = [ 1 3 x3]0 x = 1 3 x3 f (x)= x f (x) dx = [ 1 3 x3] x = 1 3 x3 1 3 3 x f (t ) dt = [ 1 3 t3] x = 1 3 x3 1 3 3 nti-derivtive by the definite integrl of f(x) x t 2 dt = 1 3 x3 + C d x dx f (t ) dt = f (x) = the Indefinite Integrl of f(x) dx = 1 3 x3 + C Intro to ODEs (0A) 20
Definite Integrls on [, ] view (I) view (II) G( x) = x 1 1 f '( x) = 1 g(x) = 1 1 dx 1 dx [ x ] = dx dy = dy dx dx view (I) view (II) f ' (x) dx [ f (x) ] = f ( ) f () g(x) dx [G(x)] = G( ) G() Intro to ODEs (0A) 21
Definite Integrls vi Indefinite Integrls nti-derivtive by the definite integrl of f(x) x f (t ) dt d x dx f (t ) dt = f (x) = the Indefinite Integrl of f(x) f ( x)dx x f (t) dt = 1 x f (t) dt + 2 f (t) dt common reference point : rbitrry [ F (x) + c ] x1 = F( ) F ( ) [ F (x) ] x1 = F ( ) F ( ) Anti-derivtive without constnt Intro to ODEs (0A) 22
Definite Integrls on [, ] view (I) view (II) G( x) = x length x 1 f '( x) = 1 g(x) = 1 re rbitrry reference point (, f()) rbitrry reference point (, F()) [ f (x) ] x1 = f ( ) f ( ) [G(x)] x1 = G( ) G( ) Intro to ODEs (0A) 23
A reference point : integrtion constnt C view (I) Anti-derivtive without constnt view (II) Anti-derivtive without constnt 1 dx f (x) = x 1 dx F (x) = x f '(x) g(x) = [ f (x)] x1 = [G(x)] x1 = {f ( ) f ()} {f ( ) f ()} = {G( ) G()} {G( ) G()} = [ f (x) f ()] x1 rbitrry reference point (, f()) = [G(x) G()] x1 rbitrry reference point (, F()) = [ f (x) + C ] x1 = [G(x) + C ] x1 = [ f ' (x)dx ] x1 = [ g(x)dx ] x1 Intro to ODEs (0A) 24
Indefinite Integrls through Definite Integrls view (I) view (II) 1 dx x 1 dx 1 dx x 1 dx f ' (x) dx f ' (x) dx g(x) dx g(x) dx = f (x) f () = x = f (x) + C = G(x) + C = G(x) G() = x f (x) = x G(x) = x Intro to ODEs (0A) 25
Definite Integrls on [, ] through F(x) Anti-derivtive (I) f ' (x) dx (II) g(x) dx d f d x dx F (x) = f (x) y = f ' (x) re f ( ) f ( ) = [ F (x)] x1 F (x) = f (x) F ( ) F ( ) length Intro to ODEs (0A) 26
Definite Integrls on [, ] nd [, ] (I) f '(x) dx (II) f (x) dx (I) f '(x) dx (II) f (x) dx Intro to ODEs (0A) 27
Definite Integrls on [, ] through F(x)+C d f d x dx F (x) = f (x) y = f ' (x) common reference point f ( ) f ( ) = [ F (x)] x1 = {F ( ) F()} {F ( ) F()} = [ F (x) F ()] x1 = [ F (x) C ] x1 = [ f (x) f ()] x1 rbitrry reference point (, F()) Intro to ODEs (0A) 28
Indefinite Integrls through F(x)+C d f d x dx = F ( x) + C x d f d x dx y = f ' (x) common reference point F (x) F () = F (x) f () f ( ) f () y = f (x) f () Intro to ODEs (0A) 29
Indefinite Integrls 1 dx x 1 dx dx dy x x + C y + C given vrible x indefinite integrl d f d x dx x c d f d x dx d f d x dx f ( ) f () f (x) f () f (x) + C given vrible x indefinite integrl Intro to ODEs (0A) 30
Derivtive Function nd Indefinite Integrls 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 = 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) 31
Intro to ODEs (0A) 32
Differentil Eqution 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) 33
First Order Exmples (y=f(x)) f ' (x) = f (x) y ' = y f (x)? y? An Exmple of A First Order Differentil Eqution f (x) = c e x for ll x y = c e x I : (, + ) f ' (x) = f (x) y ' = y f (x)? y? f (0) = 3 f (0) = 3 An Exmple of A First Order Initil Vlue Problem f (x) = 3e x for ll x y = 3e x I : (, + ) Intro to ODEs (0A) 34
Second Order Exmples (y=f(x)) f ' ' (x) = f (x) y ' ' = y f (x) = y = An Exmple of A Second Order Differentil Eqution c 1 e +x + c 2 e x for ll 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 Exmple of A Second Order Initil Vlue Problem for ll x I : (, + ) Guess the possible solution. Intro to ODEs (0A) 35
Generl First & Second Order IVPs (y=f(x)) First Order Initil Vlue 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 Initil Vlue 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) 36
References [1] http://en.wikipedi.org/ [2] M.L. Bos, Mthemticl Methods in the Physicl Sciences [3] E. Kreyszig, Advnced Engineering Mthemtics [4] D. G. Zill, W. S. Wright, Advnced Engineering Mthemtics