Constants in general solution of second order differential equations

converting1 · Dec 4, 2012

Textbook page on second-order linear differential equations: auxiliary equation am^2 + bm + c = 0, solution y = Ae^(mx) when m is a root, and general solution y = Ae^(αx) + Be^(βx) for two distinct real roots α and β, with A and B arbitrary constants.

I don't understand where the A and B come from,

if y = e^(mx), would the general solutions be y = Ae^(mx) + Be^(m_1x) assuming there are two distinct roots of the auxiliary equation? If anyone could clear this up, thanks.

dextercioby · Dec 4, 2012

A and B are the 'freedom' of the ODE, that is the maximum number of arbitrary constants which multiply the linear independent solutions. This 'freedom' is granted by the linear character of the equation (which comes from the linear characted of the differentiation operator).

In other words, if e^(alpha x) is a solution of the ODE, so is any A times e^(alpha x). The B comes from the second independent solution.

Constants in general solution of second order differential equations

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