By Brian H. Chirgwin and Charles Plumpton (Auth.)

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**Sample text**

X, y) eqn. 45) can be made to stand for more or less any first order differential equation. First we take an arbitrary point A (A'O, yo) ; this means that we do not consider a general solution but one that satisfies the 'initial' condition y = yo when x = xo. e. the integral curve in Fig. 3, as follows. , Pn, . . with It is clear that this procedure is approximate and so gives only a rough idea of the shape of the integral curve. Also the construction breaks down if a value of xn occurs for which f(xn, y„) does not exist.

We use the Taylor's series method for and find the solution for which y = 1, when x = 0. From the equation itself wefind,by putting* = 0,thsLt(dyldx)Xin,Q = 2- Hence O0)9 = 1, O^o = 2. ) It should be noticed that the method will not give the solution for values of x beyond either of the singular points. § 1 : 12] FIRST ORDER DIFFERENTIAL EQUATIONS 59 Exercises 1:11 1. The function y = y(x) satisfies the differential equation and initial condition Find the Taylor series for y in powers of x as far as the term in x4.

It is usually impracticable to take them further, either because of the unsuitable calculations involved, or because of the presence of a singularity. The methods outlined below enable the solution to be continued when once it has been started. It must be emphasised that no method is satisfactory for values of x too close to a singularity. Hence it is necessary to be able to locate the singularities and, if possible, to obtain an idea of the behaviour of the integral curves near a singularity. 2.