By Brian H. Chirgwin and Charles Plumpton (Auth.)
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This ebook first covers targeted and approximate analytical recommendations (ordinary differential and distinction equations, partial differential equations, variational ideas, stochastic processes); numerical tools (finite variations for ODE's and PDE's, finite parts, mobile automata); version inference in accordance with observations (function becoming, info transforms, community architectures, seek suggestions, density estimation); in addition to the specific position of time in modeling (filtering and country estimation, hidden Markov strategies, linear and nonlinear time series).
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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 2
15. (i) Find the orthogonal trajectories of the system of curves whose equation in plane polar coordinates is (ii) Prove that the system of curves whose equation is is auto-orthogonal, and sketch the curves of the system. § 1 : 7] FIRST ORDER DIFFERENTIAL EQUATIONS 33 1:7 Application to dynamics—resisted motion Suppose a body moves along a straight line so that at time / its displacement from a fixed point O of the line is x, its velocity is v = dx/dt and its acceleration is a = dv/dt = d2x/dt2 = v dvjdx.
Check the result by substituting this expansion in the differential equation. 5. By applying Picard's method to the equations find, as far as the term in x5, the series solution of the equation given that y = 0, dy/dx = 1 when x — 0. 6. By Picard's iterative method, or otherwise, obtain three terms of the solution of the equation dy/dx — (x2+y2)/x for which y — 0 when x = 0. 1:11 The Taylor series method If the values of the derivative of a function are known at a point, the function can be evaluated by Taylor's series at a neighbouring point, provided that the series so obtained is convergent ; this is usually true provided that the point of evaluation is not too far away from the starting point.
Therefore, when The values correct to four places are Exercises 1:10 1. Show by Picard's method that the series solution of the differential equation for which y — 0 when x = 0 is as follows : 2. By Picard's method obtain the first four non-vanishing terms of a power series in x satisfying the equation with the condition that y — 2 when x = 0. 001 of the correct value. § 1 : 11] FIRST ORDER DIFFERENTIAL EQUATIONS 57 3. 3 with that obtained in (i). 4. By Picard's method obtain an approximate solution in series, as far as the term in x7> of dy/dx = 1 - xy2, given that y = 0 when * = 0.
A Course of Mathematics for Engineers and Scientists. Volume 2 by Brian H. Chirgwin and Charles Plumpton (Auth.)