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By zalman rubinstein

ISBN-10: 1483230988

ISBN-13: 9781483230986

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Extra info for A Course in Ordinary and Partial Differential Equations

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It follows from Theorem 2 that a unique solution exists for arbitrary initial conditions and is defined for all x such that y remains finite. It should be noticed that/(x, y) does not satisfy a Lipschitz condition for all y, but only for all y in a compact set. The solutions of y' = y4/3 are y = (c — ^x)~3 and y = 0. xy-plane, there is a unique solution defined in some interval of the form (3c, oo) or ( — oo, 3c) and the solution becomes infinite as x tends to 3c. 3. EXISTENCE THEOREMS 39 (d) y' = x \y\.

We will, for convenience of notation use the symbolic notation D = d/dx, D2 = d2/dx2, etc. L(D) = Dn + aiD"-i + · · · + α„. With the operator L(D) we associate the polynomial of degree n with real coefficients φ(λ) = λ" + αίλ"-ί + . · + £,„. At this point, we shall need a few elementary results from the theory of functions of a complex variable. Lemma 3: The equation φ(λ) = 0 can be written in the form φ(λ) = (A - λν)(λ -λ2)·-·(λ- λη) = 0. (42) 61 62 ORDINARY DIFFERENTIAL EQUATIONS The Xk, k = 1, 2 , .

B) y' = y"i 0 < a < 1, where f(x, y) = ya is continuous for all (JC, y) e E2. This equation has two solutions satisfying y(0) = 0, namely y = 0 and 3. EXISTENCE THEOREMS 33 y = [(1 — α)χ] 1 / ( 1 ~ α ) . The cause for this is that the function ya does not satisfy a Lipschitz condition in an interval which contains y = 0. The solution y(x), constructed by the method of successive approxima­ tions and defined for \x — x°\

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A Course in Ordinary and Partial Differential Equations by zalman rubinstein


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