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Unformatted text preview: 18.03 Lecture #18 Oct. 19, 2009: notes Topic for today is a review of the unit. I’ll list here some of the main facts that have appeared, which you are supposed to be familiar with. An n th order differential equation (for a function y of x is in standard form if the n th derivative is expressed as a function of x and of lower-order derivatives: y ( n ) ( x ) = F ( x,y,y (1) ,y (2) ,... ,y ( n − 1) ) , (Standard n th order ODE) Here F is some function of n + 1 variables. If the derivatives are written in Leibnitz notation, the same equation looks like d n y dx n = F ( x,y, dy dx , d 2 y dx 2 ,... , d n − 1 y dx n − 1 ) , (Leibnitz n th order ODE) Solving such an equation means finding a function y that makes the equation true. This differential equation typically appears together with initial conditions : a starting point x , and n numbers: the value of y at x , and also the values of the first n − 1 derivatives at x . That is, we need to be given y ( x ) = y , y (1) ( x ) = y (1) ,... ,y ( n − 1) ( x ) = y ( n − 1) . (Initial conditions) You should know how to perform Euler’s method for finding approximate solutions to (Standard n th order ODE) and (Initial Conditions) numerically; the formulas appeared in the notes for Lecture 11. An n th order linear differential equation is 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 ( 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 “driving term” b ( x ). This linear differential equation is called homogeneous if b ( x ) = 0. We preferred to write it as y ( n ) ( x )+ a n − 1 ( x ) y ( n − 1) + ··· + a 1 ( x ) y (1) + a ( x ) y = b ( x ) (Inhomogeneous linear n th order ODE) If b ( x ) = 0, the equation is called homogeneous : y ( n ) ( x ) + a n − 1 ( x ) y ( n − 1) + ··· + a 1 ( x ) y (1) + a ( x ) y = 0 (Homogeneous n th order) Here are the big general theorems about these equations. Existence and Uniqueness Theorem. The equations (Inhomogeneous linear n th order) and (Initial conditions) have a unique solution y ( x ) , defined on any interval where the functions a i ( x ) and b ( x ) are all continuous. 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. Theorem. Suppose y p is any particular solution of (Inhomogeneous n th order). Then all solutions of (Inhomogeneous n th order) are of the form y = y p + y h , where y h is a solution of (Homogeneous n th order)....
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- Fall '09
- Derivative, nth order, homogeneous nth order