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**Unformatted text preview: **15 Homogeneous Linear Equations — Verifying the Big Theorems As promised, here we rigorously verify the claims made in the previous chapter. In a sense, there are two parts to this chapter. The first is mainly concerned with proving the claims when the differential equation in question is second order, and it occupies the first two sections. The arguments in these sections are fairly elementary, though, perhaps, a bit lengthy. The rest of the chapter deals with differential equations of arbitrary order, and uses more advanced ideas from linear algebra. If you’ve had an introductory course in linear algebra, just skip ahead to section 15.3 starting on page 323. After all, the set of differential equations of arbitrary order includes the second-order equations. If you’ve not had an introductory course in linear algebra, then you may have trouble fol- lowing some of the discussion in section 15.3. Concentrate, instead, on the development for second-order equations given in sections 15.1 and 15.2. You may even want to try to extend the arguments given in those sections to deal with higher-order differential equations. It is “do-able”, but will probably take a good deal more space and work than we will spend in section 15.3 using the more advanced notions from linear algebra. And if you don’t care about ‘why’ the results in the previous chapter are true, and are blindly willing to accept the claims made there, then you can skip this chapter entirely. 15.1 First-Order Equations While our main interest is with higher-order homogeneous differential equations, it is worth spending a little time looking at the general solutions to the corresponding first-order equations. After all, via reduction of order, we can reduce the solving of second-order linear equations to that of solving first-order linear equations. Naturally, we will confirm that our general suspicions hold at least for first-order equations. More importantly, though, we will discover a property of these solutions that, perhaps surprisingly, will play a major role is discussing linear independence for sets of solutions to higher-order differential equations. With N = 1 the generic equation describing any N th-order homogeneous linear differential equation reduces to A dy dx + By = 311 312 Homogeneous Linear Equations — Verifying the Big Theorems where A and B are functions of x on some open interval of interest I (using A and B instead of a and a 1 will prevent confusion later). We will assume A and B are continuous functions on I , and that A is never zero on that interval. Since the order is one, we suspect that the general solution (on I ) is given by y ( x ) = c 1 y 1 ( x ) where y 1 is any one particular solution and c 1 is an arbitrary constant. This, in turn, corresponds to a fundamental set of solutions — a linearly independent set of particular solutions whose linear combinations generate all other solutions — being just the singleton set { y 1 } ....

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