11 second-order ODEs

11 second-order ODEs - 18.03 Lecture #11 Oct. 2, 2009:...

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18.03 Lecture #11 Oct. 2, 2009: notes Topic for today is an introduction to higher order ordinary diFerential equations. The big topic covered in the lecture but not in the reading is Euler’s method for ±nding approximate numerical solutions. Today we put away the childish things with which we have so far been concerned, and begin to speak and reason like adults: about higher derivatives. Recall the notation y m for the m th derivative of a function y (of a single independent variable x ). An n th order ordinary diFerential equation means an equation involving the ±rst n derivatives of y , and also the independent variable x . Our experience with ±rst-order diFerential equations suggests that we can avoid some nasty behavior if we assume that our equation is solved for the highest derivative: that is, if it is in the standard form 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 ±nding a function y that makes the equation true. Example. If n = 1, the equation above becomes y = F ( x,y ), the familiar standard form for a ±rst-order ODE. In this case, in order to solve the equation, we need some additional information: an initial condition y ( x 0 ) = y 0 . This example suggests that solving (Standard n th order ODE) will also require some additional information. What additional information? The punch line is that we need to know some “starting point” x 0 , and n numbers: the value of y at x 0 , and also the values of the ±rst n - 1 derivatives at x 0 . That is, we need to be given y ( x 0 ) = y 0 , y (1) ( x 0 ) = y (1) 0 ,... ,y ( n 1) ( x 0 ) = y ( n 1) 0 . (Initial conditions) I’ll give two reasons that this is reasonable extra information to ask for. The ±rst reason is computational: using (Initial conditions), it’s possible to compute approximate numerical solutions to (Standard n th order ODE). The second reason is physical: in some large class of examples of physical systems described by a diFerential equation, (Initial conditions) is exactly what you should expect to know at the beginning in order to predict the future behavior of the system. Interlude from mathematics: in the theoretical study of diFerential equations, one seeks to prove mathematical theorems shaped like this: “there is a unique function y de±ned near x 0 and satisfying both (Standard n th order ODE) and (Initial conditions).” If you can prove such a theorem, then the conclusion is that you were asking a reasonable question. Such theorems can be motivated by examples from physics, but examples from physics can
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11 second-order ODEs - 18.03 Lecture #11 Oct. 2, 2009:...

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