Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONSwhere y 1 (x) andy 2 (x) arelinearly independent solutions of (16.1), and c 1 and c 2are constants that are fixed by the boundary conditions (if supplied).A full discussion of the linear independence of sets of functions was givenat the beginning of the previous chapter, but for just two functions y 1 and y 2to be linearly independent we simply require that y 2 is not a multiple of y 1 .Equivalently, y 1 and y 2 must be such that the equationc 1 y 1 (x)+c 2 y 2 (x) =0is only satisfied for c 1 = c 2 = 0. Therefore the linear independence of y 1 (x) andy 2 (x) can usually be deduced by inspection but in any case can always be verifiedby the evaluation of the Wronskian of the two solutions,W (x) =∣ y 1 y 2y 1 ′ y 2′ ∣ = y 1y 2 ′ − y 2 y 1. ′ (16.3)If W (x) ≠ 0 anywhere in a given interval then y 1 and y 2 are linearly independentin that interval.An alternative expression for W (x), of which we will make use later, may bederived by differentiating (16.3) with respect to x to giveW ′ = y 1 y 2 ′′ + y 1y ′ 2 ′ − y 2 y 1 ′′ − y 2y ′ 1 ′ = y 1 y 2 ′′ − y 1y ′′2 .Since both y 1 and y 2 satisfy (16.1), we may substitute for y 1 ′′ and y′′ 2 to obtainW ′ = −y 1 (py 2 ′ + qy 2 )+(py 1 ′ + qy 1 )y 2 = −p(y 1 y 2 ′ − y 1y ′ 2 )=−pW .Integrating, we find{ ∫ x}W (x) =C exp − p(u) du , (16.4)where C is a constant. We note further that in the special case p(x) ≡ 0weobtainW =constant.◮The functions y 1 =sinx and y 2 =cosx are both solutions of the equation y ′′ + y =0. Evaluate the Wronskian of these two solutions, and hence show that they are linearlyindependent.The Wronskian of y 1 and y 2 is given byW = y 1 y 2 ′ − y 2 y 1 ′ = − sin 2 x − cos 2 x = −1.Since W ≠ 0 the two solutions are linearly independent. We also note that y ′′ + y =0isa special case of (16.1) with p(x) = 0. We therefore expect, from (16.4), that W will be aconstant, as is indeed the case. ◭From the previous chapter we recall that, once we have obtained the generalsolution to the homogeneous second-order ODE (16.1) in the form (16.2), thegeneral solution to the inhomogeneous equationy ′′ + p(x)y ′ + q(x)y = f(x) (16.5)532