MethodUndeterminedCoeff

MethodUndeterminedCoeff - The Method of Undetermined...

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The Method of Undetermined Coefficients Consider the n-th order non-homogeneous equation with constant coefficients a 0 y n ( ) + a 1 y n ! 1 ( ) + ..... + a n ! 1 " y + a n y = b(x) where a 0 , a 1 , …, a n are constant and b(x) is a non-constant function of x. Let y p (x) be a particular solution of the non-homogeneous equation, containing no arbitrary constants. Let y c (x) = c 1 f 1 (x) + c 2 f 2 (x) + … + c n f n (x) where c 1 , c 2 , … , c n are arbitrary constants be the general solution of the corresponding homogeneous equation a 0 y n ( ) + a 1 y n ! 1 ( ) + ..... + a n ! 1 " y + a n y = 0 . Then, the general solution of the equation can be expressed as: y(x) = y p (x) + y c (x) In this section, we will discuss how to find the particular solution y p (x). Remark : The kind of functions b(x) for which the method of undetermined coefficients applies are actually quite restricted. Definition : A function f(x) is a UC function if it is either: 1) x n , where n is an integer n 0. 2) e ax , where a is a constant different from 0. 3) sin(bx + c), where b and c are constant and b 0. 4) cos(bx + c), where b and c are constant and b 0. 5) any function that is a finite product of two or more functions of those four types. Examples : x 3 , e -4x , sin(5x + 7), x 2 e 6x , x cos(3x), e 5x sin(3x – 7), cos(x) sin(2x), x 4 e -5x sin(2x). Remark : The method of undetermined coefficients applies when the non-homogeneous term b(x), in the non-homogeneous equation is a linear combination of UC functions. Remark : Given a UC function f(x), each successive derivative of f(x) is either itself, a constant multiple of a UC function or a linear combination of UC functions. Definition : Given a UC function f(x). We call UC set of f(x) , to the set of all UC functions consisting of f(x) itself and all linearly independent functions of which the successive derivatives of f(x) are either constant multiples or linear combinations. Example : Find the UC set of f(x) 1) Given f(x) = x 5 , its derivatives are: f’(x) = 5x 4 , f’’(x) = 20x 3 , f’’’(x) 60x 2 , f (4) = 120x, f (5) = 120, f (n) (x) = 0, n>5. The linearly independent functions of which the successive derivatives of f(x) are either constant multiples or linear combinations are: x 4 , x 3 , x 2 , x, 1. Then UC set of x 5 = {x 5 , x 4 , x 3 , x 2 , x, 1}.
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2) Given f(x) = sin 2x, its derivatives are : f’(x) = 2cos 2x, f’’(x) = -4sin 2x, f’’’(x) = -8cos 2x, f (4) = 16cos 2x they are either multiples of sin 2x or cos 2x Then, UC set of sin 2x = {sin 2x, cos 2x}. 3) Given f(x) = e ax , its derivatives are: f’(x) = ae ax , f’’(x) = a 2 e ax , …., f (n )(x) = a n e ax . They are all multiples of e
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MethodUndeterminedCoeff - The Method of Undetermined...

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