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Unformatted text preview: 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 . Its one which depends linearly on the variables y , y , . . . , y n 1 (but not necessarily linearly on x ). That is, its an equation y ( n ) ( x ) = a ( 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 , 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 ( 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 ( x ) y = 0 . (Homogeneous n th order) In either case its natural to consider this equation together with y ( x ) = y , y (1) ( x ) = y (1) , . . . , y ( n 1) ( x ) = y ( n 1) . (Initial conditions) The example that Ill 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 its not proved in 18.03. Its the one that was supposed to be motivated by the discussion in Lecture 11 of Eulers 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)....
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- Fall '09