or a form
u = (A0 + B0x) eθ x + U;
thus the general solution can be written
(A − A0)eθ x + (B − B0)eθ x + u, or {A − A0 + (B − B0)x} eθ x + u,
where A − A0, B − B0, like A, B, are arbitrary constants.
A similar result holds for a linear differential equation of any order, say
| dny | + P1 | dn-1y | + ... + Pny = R, |
| dxn | dxn-1 |
where P1, P2, ... Pn are constants, and R is a function of x. If we form the algebraic equation θn + P1θn-1 + ... + Pn = 0, and all the roots of this equation be different, say they are θ1, θ2, ... θn, the general solution of the differential equation is
y = A1 eθ1 x + A2 eθ2 x + ... + An eθn x + u,
where A1, A2, ... An are arbitrary constants, and u is any particular solution whatever; but if there be one root θ1 repeated r times, the terms A1 eθ1 x + ... + Ar eθr x must be replaced by (A1 + A2x + ... + Arxr-1)eθ1 x where A1, ... An are arbitrary constants; the remaining terms in the complementary function will similarly need alteration of form if there be other repeated roots.