Mathematical Methods for Physics and Engineering - Matematica.NET

27.7 HIGHER-ORDER EQUATIONSy1.00.80.6y0.40.20.20.40.60.8c−1.0−0.8−0.6−0.4−0.2−0.11.0 xFigure 27.6 The isocline method. The cross lines on each isocline show theslopes that solutions of dy/dx = −2xy must have at the points where theycross the isoclines. The heavy line is the solution with y(0) = 1, namelyexp(−x 2 ).second calculational point x 2 . The integration of these equations by the methodsdiscussed in the previous section presents no particular difficulty, provided thatall the equations are advanced through each particular step before any of themis taken through the following step.Higher-order equations in one dependent and one independent variable can bereduced to a set of simultaneous equations, provided that they can be written inthe formd R ydx R = f(x, y, y′ ,...,y (R−1) ), (27.82)where R is the order of the equation. To do this, a new set of variables p [r] isdefined byp [r] = dr y, r =1, 2,...,R− 1. (27.83)dxr Equation (27.82) is then equivalent to the following set of simultaneous first-orderequations:dydx = p [1],dp [r]dx = p [r+1], r =1, 2,...,R− 2, (27.84)dp [R−1]= f(x, y, p [1] ,...,p [R−1] ).dx1029