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[tex]

a\frac{d^{2}y}{dx^{2}} + b\frac{dy}{dx} + cy = 0

[/tex]

where a, b and c are constants (i deal with just constants for now, in fairness some of this stuff in discussion will translate over when they are considered as functions of x) and y is a function of x.

I am told that the general solution of this equation is of the form

[tex]

y = Au + Bv

[/tex]

where u and v are different solutions of the differential equation and A and B are non-zero constants. But surly by that that therefor implies that for every second order diff of this form there are at least 3 distinct solutions, y = u, y = v and y = Au + Bv.

Looking around the net and in books I don't see why this has to be so. I am either given examples and then from that it is implied this must be so, or even worse it is simply stated, I find this extremely frustrating as there is no rigor in any of the stuff I have read. Could anyone provide me with information on either why this is so, or links to more in depth information. Thanks everyone :-)