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Unformatted text preview: 20 Nonhomogeneous Equations in General Now that we know how to solve a couple of rather broad classes of homogeneous equations, it is time to start looking at nonhomogeneous equations. 20.1 Basic Theory Recollections about Linearity ∗ Let us go back to our generic, N thorder, linear differential equation, a d N y dx N + a 1 d N − 1 y dx N − 1 + · · · + a N − 2 d 2 y dx 2 + a N − 1 dy dx + a N y = g . Remember, g and the a k ’s denote known functions of x over some interval of interest, I . As usual, we will assume these functions are continuous and that a is never zero on this interval. As before, it is convenient to let L denote the corresponding differential operator from the left side of the equation, L = a d N dx N + a 1 d N − 1 dx N − 1 + · · · + a N − 2 d 2 dx 2 + a N − 1 d dx + a N . That is, given any sufficiently differentiable function φ( x ) on I , L [ φ ] = a d N φ dx N + a 1 d N − 1 φ dx N − 1 + · · · + a N − 2 d 2 φ dx 2 + a N − 1 d φ dx + a N φ . Using this operator, we can write our generic differential equation as L [ y ] = g . Remember, this equation is said to be homogeneous if g is always zero on our interval, and nonhomogeneous otherwise. Since we have already discussed the homogeneous case, let us now assume g ( x ) is a function that is nonzero over at least a portion of our interval of interest. Don’t forget, however, that, for each nonhomogeneous equation L [ y ( x ) ] = g ( x ) , ∗ You may want to briefly review the material in chapter 12. 409 410 Nonhomogeneous Equations in General we still have the corresponding homogeneous equation L [ y ( x ) ] = where we simply replace g ( x ) with 0 . This equation will play a significant role in solving the nonhomogeneous equation. General Solutions to Nonhomogeneous Equations Recall that, if we have a bunch of sufficiently differentiable functions — φ 1 ( x ) , φ 2 ( x ) , …, φ K ( x ) (where K is some positive integer) — and a corresponding set of constants — c 1 , c 2 , …, c K — then L [ c 1 φ 1 ( x ) + c 2 φ 2 ( x ) + · · · + c K φ K ( x ) ] = c 1 L [ φ 1 ( x ) ] + c 2 L [ φ 2 ( x ) ] + · · · + c K L [ φ K ( x ) ] . We used this to construct general solutions to homogeneous equations as linear combinations of different solutions. With non homogeneous equations we must be a little more careful. After all, if y p ( x ) and y q ( x ) are two particular solutions to a nonhomogeneous equation L [ y ( x ) ] = g ( x ) (i.e., L [ y p ( x ) ] = g ( x ) and L [ y q ( x ) ] = g ( x ) ), and c 1 and c 2 are any two constants that do not add up to 1 , then, since g is a nonzero function, L [ c 1 y p ( x ) + c 2 y q ( x ) ] = c 1 L [ y p ( x ) ] + c 2 L [ y q ] = c 1 g ( x ) + c 2 g ( x ) = [ c 1 + c 2 ] g ( x ) negationslash= g ( x ) ....
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 Summer '09
 Differential Equations, Equations, Derivative, 1 L, 2 L, Nonhomogeneous Equations

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