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c02 - 18.03 Class 2 Numerical Methods[1 The study of...

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18.03 Class 2, February 10, 2006 Numerical Methods [1] The study of differential equations rests on three legs: . Analytic, exact, symbolic methods . Quantitative methods (direction fields, isoclines .... ) . Numerical methods Even if we can solve symbolically, the question of computing values remains. The number e is the value y(1) of the solution to y' = y with y(0) = 1. But how do you find that in fact e = 2.718282828459045 .... ? The answer is: numerical methods. As an example, take the first order ODE y' = x - y^2 = F(x,y) with initial condition y(0) = 1. Question: what is y(1) ? I revealed a picture of the direction field with this solution sketched. This is what we had on Wednesday but upside down: then we considered y' = y^2 - x . The funnel is at the top this time. This solution seems to be one of those trapped in the funnel, so for large x , the graph of y(x) is close to the graph of \sqrt (x) . But what about y(1) ? Here's an approach: use the tangent line approximation! Since F(0,1) = -1 , the straight line best approximating the integral curve at (0,1) has slope -1 , and goes through the point (0,1) : so this gives the estimate y(1) is approximately 0 . Well, we know that the integral curve is NOT straight. What to do? Approximate it by a polygon! So use the tangent line approximation to go half way, and then check the direction field again: y(.5) is approximately 1 + (slope)(run) = 1 + (-1)(.5) = .5 (.5,.50) is a vertex on this polygon. The direction field there has slope F(.5,.5) = .5 - (.5)^2 = .25 .
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