Mathematical Methods for Physics and Engineering - Matematica.NET

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONSIf the original equation (15.1) has f(x) = 0 (i.e. it is homogeneous) then ofcourse the complementary function y c (x) in (15.3) is already the general solution.If, however, the equation has f(x) ≠ 0 (i.e. it is inhomogeneous) then y c (x) isonlyone part of the solution. The general solution of (15.1) is then given byy(x) =y c (x)+y p (x), (15.7)where y p (x)istheparticular integral,whichcanbeany function that satisfies (15.1)directly, provided it is linearly independent of y c (x). It should be emphasised forpractical purposes that any such function, no matter how simple (or complicated),is equally valid in forming the general solution (15.7).It is important to realise that the above method for finding the general solutionto an ODE by superposing particular solutions assumes crucially that the ODEis linear. For non-linear equations, discussed in section 15.3, this method cannotbe used, and indeed it is often impossible to find closed-form solutions to suchequations.15.1 Linear equations with constant coefficientsIf the a m in (15.1) are constants rather than functions of x then we haved n ya ndx n + a d n−1 yn−1dx n−1 + ···+ a dy1dx + a 0y = f(x). (15.8)Equations of this sort are very common throughout the physical sciences andengineering, and the method for their solution falls into two parts as discussedin the previous section, i.e. finding the complementary function y c (x) and findingthe particular integral y p (x). If f(x) = 0 in (15.8) then we do not have to finda particular integral, and the complementary function is by itself the generalsolution.15.1.1 Finding the complementary function y c (x)The complementary function must satisfyd n ya ndx n + a d n−1 yn−1dx n−1 + ···+ a dy1dx + a 0y = 0 (15.9)and contain n arbitrary constants (see equation (15.3)). The standard methodfor finding y c (x) is to try a solution of the form y = Ae λx , substituting this into(15.9). After dividing the resulting equation through by Ae λx , we are left with apolynomial equation in λ of order n; thisistheauxiliary equation and readsa n λ n + a n−1 λ n−1 + ···+ a 1 λ + a 0 =0. (15.10)492