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