By Gianni Dal Maso
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Additional info for An introduction to G-convergence
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.
An introduction to G-convergence by Gianni Dal Maso