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lecture_11 (dragged) 2

lecture_11 (dragged) 2 - Then we have dy dt = lim n →∞...

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MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 6 3 Unfortunately, this is not entirely useful because both sides of the equation involve y = y ( t ). We use this idea to define a sequence of functions y n = y n ( t ) which will approximate the exact solution y = y ( t ). First define the constant function y 0 ( t ) = y 0 . Now define recursively the sequence of functions y n +1 ( t ) = t t 0 G τ , y n ( τ ) d τ + y 0 . Note that d dt y n +1 = G t, y n and y n +1 ( t 0 ) = y 0 . Now consider the limit: y ( t ) = lim n →∞ y
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Unformatted text preview: ) . Then we have dy dt = lim n →∞ d dt ³ y n +1 ´ = lim n →∞ G ± t, y n ² = G ( t, y ) , y ( t ) = lim n →∞ y n ( t ) = lim n →∞ y = y . Hence y = y ( t ) is the exact solution to the initial value problem. This technique of approximating the solution y = y ( t ) via a sequence y n = y n ( t ) is known as Picard’s Method ....
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