Consider the differential equation. `Dy/dx +P(x)y = Q(x)` (i) If two particular solutions of given equation u(x) and v(x) are known, find the general solution of the same equation in terms of u(x) and v(x). (ii) If `alpha and beta` are constants such that the linear combinations `alpha.u(x) + beta.v(x)` is a solution of the given equation, find the relation between `alpha and beta`. (iii) If w(x) is the third particular solution different from u(x) and v(x) then find the ratio v(x) u(x)/w(x)-u(x)

Answer

(i) y = u(x) + K(u(x) – v(x)) where K is any constant ; (ii) `alpha + beta` = 1; (iii) constant