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## Main Question or Discussion Point

I have this problem:

Consider the differential equation

y'' + P(x) y' + Q(x) y = 0

on the interval a ≤ x ≤ b. Suppose we know two solutions y

y

y

Give the solution of the equation

y'' + P(x) y' + Q(x) y = f(x)

which obeys the conditions y(a) = y(b) = 0 in the form

[PLAIN]http://mathbin.net/equations/57612_0.png [Broken]

where G(x,x') involves only the solutions y

Ok, I don't really know where to begin here. My initial reaction is that I need to use variation of parameters, because it's the only method I know of that's going to give me a solution in terms of an integral, but I'm not sure if that makes sense. Anyway, if I do that, I get an expression like

[PLAIN]http://mathbin.net/equations/57613_0.png [Broken]

which I don't think is right because it doesn't obey the y(a) = y(b) = 0 condition. I also don't get that "assumes different functional forms..." part. I really have no idea what I'm doing here, I'm mostly flailing aimlessly.

Consider the differential equation

y'' + P(x) y' + Q(x) y = 0

on the interval a ≤ x ≤ b. Suppose we know two solutions y

_{1}(x), y_{2}(x) such thaty

_{1}(a) = 0, y_{1}(b) ≠ 0y

_{2}(a) ≠ 0, y_{2}(b) = 0Give the solution of the equation

y'' + P(x) y' + Q(x) y = f(x)

which obeys the conditions y(a) = y(b) = 0 in the form

[PLAIN]http://mathbin.net/equations/57612_0.png [Broken]

where G(x,x') involves only the solutions y

_{1}, y_{2}and assumes different functional forms for x' < x and x' > x.Ok, I don't really know where to begin here. My initial reaction is that I need to use variation of parameters, because it's the only method I know of that's going to give me a solution in terms of an integral, but I'm not sure if that makes sense. Anyway, if I do that, I get an expression like

[PLAIN]http://mathbin.net/equations/57613_0.png [Broken]

which I don't think is right because it doesn't obey the y(a) = y(b) = 0 condition. I also don't get that "assumes different functional forms..." part. I really have no idea what I'm doing here, I'm mostly flailing aimlessly.

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