Download e-book for iPad: An introduction to G-convergence by Gianni Dal Maso

By Gianni Dal Maso

ISBN-10: 081763679X

ISBN-13: 9780817636791

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Existence and Uniqueness of Solution Theorem 1. Consider the initial value problem dy ϭ f A x, y B , dx y A x0 B ϭ y0 . If f and 0f / 0y are continuous functions in some rectangle R ϭ E A x, y B : a 6 x 6 b, c 6 y 6 dF that contains the point A x0, y0 B , then the initial value problem has a unique solution f A x B in some interval x0 Ϫ d Ͻ x Ͻ x0 ϩ d, where d is a positive number. 5 Layout for the existence–uniqueness theorem The preceding theorem tells us two things. First, when an equation satisfies the hypotheses of Theorem 1, we are assured that a solution to the initial value problem exists.

That is, we solve the initial value problem m dy ϭ mg Ϫ by , dt y(0) ϭ y0 . Substituting y ϭ y0 and t ϭ 0 into the general solution to the differential equation, we can solve for A. With this value for A, the solution to the initial value problem is (5) Y‫؍‬ mg mg ؊bt m ؉ aY0 ؊ be / . 2 Graph of y A t B for six different initial velocities y0. 8 m/sec2, m / b ϭ 5 sec) The preceding formula gives the velocity of the object falling through the air as a function of time if the initial velocity of the object is y0.

Dy / dx ϭ x / y , 2 y A0B ϭ 0 15. dy / dx ϭ 2x Ϫ y , 16. 4 17. From a sketch of the direction field, what can one say about the behavior as x approaches ϩq of a solution to the following? 3xy dy ‫ ؍‬2 dx 2x ؊ y2 and the equipotential lines satisfy the equation 18. From a sketch of the direction field, what can one say about the behavior as x approaches ϩq of a solution to the following? dy ϭ Ϫy dx 19. By rewriting the differential equation dy / dx ϭ Ϫy / x in the form Ϫ1 1 dy ϭ dx y x integrate both sides to obtain the solution y ϭ C / x for an arbitrary constant C.

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An introduction to G-convergence by Gianni Dal Maso


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