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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 th-order, 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 ( x ) on I , 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 . 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. Dont 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 1 ( x ) , 2 ( x ) , , K ( x ) (where K is some positive integer) and a corresponding set of constants c 1 , c 2 , , c K then L [ c 1 1 ( x ) + c 2 2 ( x ) + + c K K ( x ) ] = c 1 L [ 1 ( x ) ] + c 2 L [ 2 ( x ) ] + + c K L [ 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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