New PDF release: Abdus Salam - Weak and Electromagnetic Interactions.

By Nils Fridolf Valdemar Svartholm

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Extra info for Abdus Salam - Weak and Electromagnetic Interactions. Published in Elementary Particle Theory: Proceedings of the 8th Nobel Symposium

Sample text

Construct a figure illustrating the √ fact that the initial value problem y = 2 y, y(0) = 0 has infinitely many different solutions. (b) For what values of √ b does the initial value problem y = 2 y, y(0) = b have (i) no solution, (ii) a unique solution that is defined for all x? 28. Verify that if k is a constant, then the function y(x) ≡ kx satisfies the differential equation x y = y for all x. Construct a slope field and several of these straight line solution curves. Then determine (in terms of a and b) how many different solutions the initial value problem x y = y, y(a) = b has—one, none, or infinitely many.

Y(x) = + 4 − x 2 and y(x) = − 4 − x 2 satisfies the initial condition y(0) = 2 (Fig. 4). Remark 1: You should not assume that every possible algebraic solution y = y(x) of an implicit solution satisfies the same differential equation. For instance, if we multiply the implicit solution x 2 + y 2 − 4 = 0 by the factor (y − 2x), then we get the new implicit solution (y − 2x)(x 2 + y 2 − 4) = 0 that not only the previously noted explicit solutions y = √ √ yields (or “contains”) 2 + 4 − x and y = − 4 − x 2 of the differential equation x + yy = 0, but also the additional function y = 2x that does not satisfy this differential equation.

2) is given explicitly by y(x) = Cekx . A slope field suggests visually the general shapes of solution curves of the differential equation. Through each point a solution curve should proceed in such a direction that its tangent line is nearly parallel to the nearby line segments of the slope field. Starting at any initial point (a, b), we can attempt to sketch freehand an approximate solution curve that threads its way through the slope field, following the visible line segments as closely as possible.

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Abdus Salam - Weak and Electromagnetic Interactions. Published in Elementary Particle Theory: Proceedings of the 8th Nobel Symposium by Nils Fridolf Valdemar Svartholm


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