Download e-book for iPad: A Course in Higher Mathematics Volume II: Advanced Calculus by Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith

By Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith

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Extra resources for A Course in Higher Mathematics Volume II: Advanced Calculus

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YP(x) = f(x). (19) We introduce a new required function z in place of y, given by: V = Vi (x) + z. Substitution in (18) gives us the equation for z: or, using identity (19): 2/ln) + *

22 ORDINARY DIFFERENTIAL EQUATIONS [7 x* xy y X *--£■*! 2244 The results of the computation are shown in the accompanying table. e. xy/2, the fourth the difference Ay = ys+x — ys, and the last the value of the ordinate of the accurate integral curve y = ex2/4. e. 5%. 7. The general solution. On altering the value of y in the initial condition: we obtain an infinite set of solutions of equation (42), or in geometrical terms, a family of integral curves depending on the arbitrary constant y0, this being the ordinate of the point of intersection of an integral curve with the line x = x0.

The orthogonal trajectories of the parabolas y = Cx2 are illustrated in Fig. 14. § 2. Differential equations of higher orders; systems of equations 13. , y(n)) = o, or, on solving with respect to (i) y^: = / ( * , y, y', y", . . , y(»~»). (2) Every function y of the independent variable x t h a t satisfies equation (1) or (2) is called a solution of the equation, whilst the actual task of finding the solutions of the equation is described as the task of integrating the equation. We take as an example the linear motion of a point-mass of mass m under the action of a force F, which depends on time t, on the position of the point and on its velocity.

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A Course in Higher Mathematics Volume II: Advanced Calculus by Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith


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