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The problem as to whether this iterative process is or is not convergent requires examination. It is shown in books on analysis that this process is convergent if a domain D can be found in the x-y plane which includes the point (x0, yo) and in which/(x, y) is single-valued and continuous, and if for any two points (x, y{) and (x, y2) on the same arbitrary ordinate inside D the condition where K is a constant, is satisfied. (This is called a Lipschitz condition) Example 1. Use Picard's method to obtain the solution in powers of x as far as the term in x7 of the differential equation where y = 0 when x = 0.

Successive differentiations of a differential equation often permit us to evaluate the derivatives for convenient starting values of x. Example 1. We consider the simple equation and obtain the solution for which y = 1 when x — 0. By successive differentiations we obtain 58 A COURSE OF MATHEMATICS Here we have denoted (dnyldxn)x^0 by (yn)0. Hence Taylor's theorem gives This is, of course, the well known series for ckx and is convergent for all values of x and so we have found the solution for all values of x.

Assuming the power P of the engine to be constant, show that he speed v of the train at time / after the start of the motion is given by The power is cut off when the train has speed V and the train is stopped in a further time T by means of a constant braking force F. Show that The acceleration a is given by (1) To find the v-t relation we write a — dv/dt and eqn. e. (2) Since k =* 0, eqn. e. the train has a terminal velocity U to which v tends asymptotically as t increases. This result also follows from eqn.

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