Math118_S10_W5

Math118_S10_W5 - Math118 (M. Kohandel) -Outline (week 5)...

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 5) 5.1 Reducible second-order DEs (15.4) Dependent variable missing Independent variable missing 5.2 Taylor polynomials, remainders, and series (10.3) Taylor polynomials Taylor's reminder formula Maclaurin series 5.3 Power series (10.4) Power series Radius of convergence Sums of power series
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 5.1 Reducible second-order DEs Second-order DEs in y (dependent variable) as a function of x (independent variable) can be symbolically expressed as 0 ) , , , ( 2 2 x y dx dy dx y d f (1) If the dependent variable does not appear explicitly in a second-order DE, then it can be reduced to a first order DE using the substitution dx dy v / : 0 ) , , ( ) , , ( , 2 2 2 2 x v dx dv f x dx dy dx y d f v dx dy dx dv dx y d This gives ) ( x v , which can be used to obtain ) ( x y . (2) If the independent variable does not appear explicitly in a second-order DE, then it can be reduced to a first order DE using the substitution dx dy v / and nothing that: 0 ) , , ( ) , , ( , 2 2 2 2 y v dy dv v f y dx dy dx y d f v dx dy dy dv v dx dy dy dv dx dv dx y d This gives ) ( y v , which can be used to obtain ) ( x y . Example 5.1: Solve the following ODEs. 2 1 1 1 1 / 2 2 ) 1 ( 1 1 1 | 1 | ln 1 1 1 ) a ( C e C x dx e C y e C dx dy e C v e v c x v dx v dv v dx dv dx dy dx y d x x x x c x dx dy v  )] ( 2 tan[ 2 2 tan 2 2 1 2 1 2 1 ) b ( 2 2 1 2 2 2 / 2 2 C t C C x C t C x C dt C x dx C x dt dx C x v xdx dv xv dx dv v dt dx x dt x d dt dx v 5.2 Taylor polynomials, remainders, and series Recall from Math 116 that the equation of tangent line to the curve of ) ( x f at a point )) ( , ( 0 0 x f x can be obtained by considering the secant line joining )) ( , ( 0 0 x f x to a second point )) ( , ( 0 0 x x f x x
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 3 and letting 0 x . In the linear approximation, we use this tangent line as an approximation to ) ( x f for values of x close to 0 x . ) )( ( ) ( ) ( : ion approximat Linear ) )( ( ) ( : line tangent of Equation ) ( ) ( ) ( ) ( : line secant of Equation 0 0 0 0 0 0 0 0 0 0 0 x x x f x f x f x x x f x f y x x x x f x x f x f y x  We know that for any set of 1 n equidistance points, a polynomial of degree n can be found which passes through all of them (recall that this idea was used in Simpson's rule). Hence, to generalize the linear approximation, we can use three points and find the equation of the second degree polynomial (parabola) which passes through them: If we substitute the three points ) 2 ( , ) ( , ) ( 2 , , 0 2 0 1 0 0 0 2 0 1 0 0 x x f y x x f y x f y x x x x x x x x into the equation of parabola 2 cx bx a y , we obtain 2 2 0 2 0 2 0 2 0 2 1 2 2 2 0 1 0 2 2 1 2 2 1 2 1 2 1 2 0 2 1 0 1 0 1 0 2 2 2 2 2 1 1 1 2 0 0 0 ) ( 2 ] ) ( 2 ) 2 [( ) 2 ( ) ( ) ( , ) ( ) ( , , 0 1 2 x c x x x x x c x x x c y y y x x c x x b y y y
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This note was uploaded on 02/06/2011 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.

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Math118_S10_W5 - Math118 (M. Kohandel) -Outline (week 5)...

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