Mathematical Methods for Physics and Engineering - Matematica.NET

14.2 FIRST-DEGREE FIRST-ORDER EQUATIONSare several different types of first-degree first-order ODEs that are of interest inthe physical sciences. These equations and their respective solutions are discussedbelow.14.2.1 Separable-variable equationsA separable-variable equation is one which may be written in the conventionalformdy= f(x)g(y), (14.3)dxwhere f(x) andg(y) are functions of x and y respectively, including cases inwhich f(x) org(y) is simply a constant. Rearranging this equation so that theterms depending on x and on y appear on opposite sides (i.e. are separated), andintegrating, we obtain∫ ∫ dyg(y) = f(x) dx.Finding the solution y(x) that satisfies (14.3) then depends only on the ease withwhich the integrals in the above equation can be evaluated. It is also worthnoting that ODEs that at first sight do not appear to be of the form (14.3) cansometimes be made separable by an appropriate factorisation.◮Solvedydx = x + xy.Since the RHS of this equation can be factorised to give x(1 + y), the equation becomesseparable and we obtain∫ ∫dy1+y = xdx.Now integrating both sides separately, we findln(1 + y) = x22 + c,and so( ) x21+y =exp2 + c = A expwhere c and hence A is an arbitrary constant. ◭Solution method. Factorise the equation so that it becomes separable. Rearrangeit so that the terms depending on x and those depending on y appear on oppositesides and then integrate directly. Remember the constant of integration, which canbe evaluated if further information is given.( x22),471