L63 - Spring 2003 Math 308/501502 6 Theory of Higher-Order...

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Spring 2003 Math 308/501–502 6 Theory of Higher-Order Linear ODEs 6.3 Method of Undetermined Coeffs Wed, 01/Oct c ± 2003, Art Belmonte Summary A nonhomogeneous linear ODE of order n with real constant coefficients has the form L [ y ] = n X k = 0 a k y ( k ) = f .Here t (or x )is the independent variable and the nonhomogeneity f 6= 0 is known as the “forcing function;” it may consist of one or several terms. The associated homogeneous ODE is L [ y ] = n X k = 0 a k y ( k ) = 0. A general solution y h of this is obtained via §6 . 2 methods. General solution of the nonhomogeneous equation If y p is a particular solution of the nonhomogeneous equation and y h is a general solution to the associated homogeneous equation, then a general solution of the nonhomogeneous equation is given by y = y p + y h . Superposition Principle For k = 1 , 2 ,..., M ,let y p k be a solution of L [ y ] = f k . Then for any constants c 1 c M , the function y p = M X k = 1 c k y p k solves the differential equation L [ y ] = M X k = 1 c k f k . (This follows immediately from the fact that L is a linear differential operator.) Method of Undetermined Coefficients If the forcing function f ( t ) = M X k = 1 f k ( t ) is a sum of products of real polynomials, sines, cosines, and/or exponentials, then the following method produces a general solution of L [ y ] = f ( t ) . Each f k ( t ) must have the form e α t ( p u ( t ) cos β t + q v ( t ) sin β t ) where p u and q v are polynomials of degrees u and v , whereas α and β are real constants. Now α or β may be 0, p u or q v may be constants, and these may all vary with the index k .Of ten , M is 1; i.e., there is a single term in the forcing function. 1. First obtain a general solution y h of L [ y ] = 0. 2. For each k , determine a particular solution y p k of L [ y ] = f k ( t ) , as follows. (If M = 1, just use y p for y p 1 .) (a) Form y p k = t s e α t ( P N ( t ) cos β t + Q N ( t ) sin β t ) , where N = max ( u ,v) and s is the smallest nonnegative
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L63 - Spring 2003 Math 308/501502 6 Theory of Higher-Order...

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