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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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