Download e-book for kindle: 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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N + 1) an+1 xn + ... + an_t an~* - f . . ) . We equate coefficients of like powers of x on both sides and obtain the relationships shown in the table. It is clear from these that a i = «3 = a 5 = • • • = a2/H 1 = . . e. + n! a (w + i ) % i = y a n-i 4 6. The Euler—Cauchy method. The approximate construction for an integral curve of equation (42) given in the previous article can be simplified b y using lines only, parallel to O F , instead of the mesh of squares. This modified Eulerian method results in a relatively simple and handy means of evaluating approximately the ordinate y of an integral curve for a previously assigned abscissa x.

Are evidently independent of the choice of scale for the intercepts. This construction makes it clear by inspection t h a t one and only one integral curve of equation (42) passes through a given point (*o> Vo)This assertion is susceptible of rigorous proof if the function f(x, y) has properties in addition to continuity. For instance, if f(x, y) is a single-valued, continuous function of its arguments in the neighbourhood of the point (xQ) y0) and has a continuous derivative with respect to y, one and only one integral curve of equation (42) passes through the point {xQ, y0).

V . . are evidently independent of the choice of scale for the intercepts. This construction makes it clear by inspection t h a t one and only one integral curve of equation (42) passes through a given point (*o> Vo)This assertion is susceptible of rigorous proof if the function f(x, y) has properties in addition to continuity. For instance, if f(x, y) is a single-valued, continuous function of its arguments in the neighbourhood of the point (xQ) y0) and has a continuous derivative with respect to y, one and only one integral curve of equation (42) passes through the point {xQ, y0).

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


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