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Unformatted text preview: 18.03 Lecture #8 Sept. 25, 2009: notes The topic today is autonomous differential equations . The definition is simple: they are equa- tions of the form dy dt = f ( y ) . (Autonomous) I’ve called the independent variable t instead of x because in many examples such an equation describes the change of a variable y with time. The word “autonomous” means that y controls itself; in order to know how to change, y only needs to know its own value, and not what time it is. An autonomous equation therefore describes a system in which the rules don’t change with time. An autonomous equation is always separable. As long as f ( y ) is not zero, you can write it as dy f ( y ) = dt, and get an implicit description of the general solution integraldisplay dy f ( y ) = t + C. As we’ll see in some examples, even a harmless-looking function f can make this integral difficult to evaluate explicitly; and even if you can do that, solving for y as a function of t can be painful. Here’s an example of these difficulties. Start with y ′ = cos( y ) , (Cosine Autonomous) To make a physical model, think of a grocery store conveyor belt moving at a constant speed of 1. Over the middle of the belt suspend a (frictionless) axle pointed at the belt, with a rod (of length 1) that can make a complete circle over the belt. At the end of the rod attach a little unicycle-mounted action figure that rides on the belt. The belt tries to carry the unicycle in a straight line, but the attached rod holds it in a circle. You can let y represent the angular position of the rod, with y = 0 perpendicular to the motion of the belt. Then (Cosine Autonomous) describes the angular velocity of the unicycle: it’s the component of the belt’s velocity tangent to the circle in which the unicycle must move. Physically it’s clear what should happen. If the rod starts out parallel to the motion of the belt, then it stays in that position; this equilibrium is stable at the top ( y = π/ 2 plus a multiple of 2 π ) and unstable at the bottom ( y = − π/ 2 plus a multiple of 2 π ). If the rod starts to the left of vertical ( y between − π/ 2 and π/ 2 plus a multiple of 2 π ) then it’s swept up toward the top on the left, never quite reaching it. If it starts to the right of vertical ( y between − π/ 2 and − 3 π/ 2 plus a multiple of 2 π ) then it’s swept up toward the top on the right, again never quite reaching it. So let’s see how to solve analytically. We already know about the constant solutions y = (odd multiple of π/ 2), so we can assume that cos( y ) is not zero (and therefore that sin( y ) negationslash = ± 1). The equation (Cosine Autonomous) is sec( y ) dy = dt , which integrates to ln | sec( y ) + tan( y ) | = t + C, (1 + sin( y )) / cos( y ) = Ae t ....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Fall '09 term at MIT.
- Fall '09