03 linear first-order ODEs, integrating factors, models

03 linear first-order ODEs, integrating factors, models -...

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Unformatted text preview: 18.03 Lecture #3 Sept. 14, 2009: notes Big topic today is how to solve a special kind of first-order differential equations called linear equations. These are the ones of the form dy dx =- P ( x ) y + Q ( x ) , ( LinearODE ) y ( x ) = y ( IC ) Theorem. Suppose that the functions P ( x ) and Q ( x ) are continuous on an interval a ≤ x ≤ b , and suppose also that x belongs to this interval: a ≤ x ≤ b . Then there is a unique solution y ( x ) to (LinearODE), (IC) defined on the interval a ≤ x ≤ b . It can be found by computing integrals: y ( x ) = e − integraltext x x P ( s ) ds bracketleftbiggintegraldisplay x x parenleftbigg Q ( t ) e integraltext t x P ( s ) ds parenrightbigg dt + y bracketrightbigg . You should (first) not memorize the formula for the solution in this theorem, but instead memorize the proof —that is, the procedure that leads to the formula—sketched below. It’s easier and more reliable to repeat the procedure in each problem. Before the proof, general comments. This theorem is shaped like the one about the general ODE dy dx = f ( x,y ) ( ODE ) in the last lecture. The hypotheses are much narrower: instead of allowing any continuous function of x and y , we allow only functions that are linear in the variable y . What we gain from these narrow hypotheses is two things. First, the conclusion is stronger: the solution exists on the whole interval where the equation is defined, and not just on some (hard to identify) smaller subinterval. Second, we get an integral formula for the solution. Always this formula can be computed numerically to whatever accuracy we want. Often we can compute the antiderivatives in closed form and get an elementary formula for the solution. The last comment is to address the question ... who cares? Why should we spend effort on such a very special kind of differential equation? (You may particularly ask such a question after seeing the dozens of problems about tanks of brine in EP; is mixing salt solutions worth so much time?) One kind of answer is that Course 6 is full of linear differential equations. The basic laws governing the behavior of electrical circuits can be expressed as linear differential equations, for...
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03 linear first-order ODEs, integrating factors, models -...

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