Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODS(ii) To order h 4 ,wherey i+1 = y i + 1 6 (c 1 +2c 2 +2c 3 + c 4 ), (27.80)c 1 = hf(x i ,y i ),c 2 = hf(x i + 1 2 h, y i + 1 2 c 1),c 3 = hf(x i + 1 2 h, y i + 1 2 c 2),c 4 = hf(x i + h, y i + c 3 ).27.6.5 IsoclinesThe final method to be described for first-order differential equations is not somuch numerical as graphical, but since it is sometimes useful it is included here.The method, known as that of isoclines, involves sketching for a number ofvalues of a parameter c those curves (the isoclines) in the xy-plane along whichf(x, y) =c, i.e. those curves along which dy/dx is a constant of known value. Itshould be noted that isoclines are not generally straight lines. Since a straightline of slope dy/dx at and through any particular point is a tangent to the curvey = y(x) at that point, small elements of straight lines, with slopes appropriateto the isoclines they cut, effectively form the curve y = y(x).Figure 27.6 illustrates in outline the method as applied to the solution ofdy= −2xy. (27.81)dxThe thinner curves (rectangular hyperbolae) are a selection of the isoclines alongwhich −2xy is constant and equal to the corresponding value of c. The smallcross lines on each curve show the slopes (= c) that solutions of (27.81) musthave if they cross the curve. The thick line is the solution for which y =1atx = 0; it takes the slope dictated by the value of c on each isocline it crosses. Theanalytic solution with these properties is y(x) =exp(−x 2 ).27.7 Higher-order equationsSo far the discussion of numerical solutions of differential equations has beenin terms of one dependent and one independent variable related by a first-orderequation. It is straightforward to carry out an extension to the case of severaldependent variables y [r] governed by R first-order equations:dy [r]dx = f [r](x, y [1] ,y [2] ,...,y [R] ), r =1, 2,...,R.We have enclosed the label r in brackets so that there is no confusion between,say, the second dependent variable y [2] and the value y 2 of a variable y at the1028