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13 thms about higher-order ODEs

13 thms about higher-order ODEs - 18.03 Lecture#13 Oct 7...

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18.03 Lecture #13 Oct. 7, 2009: notes Today is a bit of a grab bag: some extremely general theorems about solutions of linear dif- ferential equations, and some rather specific methods for finding and working with those solutions. Recall from Lecture 11 the general definition of an n th order linear differential equation . It’s one which depends linearly on the variables y , y , . . . , y n 1 (but not necessarily linearly on x ). That is, it’s an equation y ( n ) ( x ) = a 0 ( x ) y a 1 ( x ) y (1) a 2 ( x ) y (2) − · · · − a n 1 ( x ) y ( n 1) + b ( x ) . (Linear n th order ODE) Here the coefficient functions a 0 , a 1 , . . . , a n 1 can be any functions of x , as can the “constant term” b ( x ). This linear differential equation is called homogeneous if b ( x ) = 0. If we use the notation D = d dx , then this equation can be written as D n y + a n 1 ( x ) D n 1 y + · · · + a 1 ( x ) Dy + a 0 ( x ) y = b ( x ) . (Inhomogeneous n th order) The homogeneous version is D n y + a n 1 ( x ) D n 1 y + · · · + a 1 ( x ) Dy + a 0 ( x ) y = 0 . (Homogeneous n th order) In either case it’s natural to consider this equation together with y ( x 0 ) = y 0 , y (1) ( x 0 ) = y (1) 0 , . . . , y ( n 1) ( x 0 ) = y ( n 1) 0 . (Initial conditions) The example that I’ll often refer to is m d 2 y dt 2 + c dy dt + ky = b ( t ) . (Forced damped spring normalized) I explained last time that c refers to the damping action of a “dashpot,” which exerts a decelerating force proportional to the velocity; k is the spring constant; and the new term b ( t ) is an externally applied driving force applied to the mass. Here are the big general theorems about these equations. Existence and Uniqueness Theorem. The equations (Inhomogeneous n th order) and (Ini- tial conditions) have a unique solution y ( x ) , defined on any interval where the functions a i ( x ) and b ( x ) are all continuous. (This is a hard theorem, and it’s not proved in 18.03. It’s the one that was supposed to be motivated by the discussion in Lecture 11 of Euler’s method, which gives a method for constructing approximations to the solution. In the case n = 1, you learned a method for computing the solution by doing integrals. For higher n there is no generalization of that method that always works.) Superposition Theorem. Suppose y 1 and y 2 are any solutions of (Homogeneous n th order). Then c 1 y 1 + c 2 y 2 is again a solution. (This is a very easy theorem, which earns the title “theorem” because it’s so useful.) Everything else is a consequence of these two fundamental theorems. Here are some general- izations of what you learned about first-order linear ODE.
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