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# The homogeneous equation....how to solve? might require the use of greens function? watch

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1. The homogeneous equation:

y''[x] + p[x]y'[x] + q[x]y[x] = 0

is subject to the inhomogeneous boundary conditions
y[a] = (alpha)
y[b] = (beta)
considering a transformation y[x] = Y[x] + P[x],
where P[x] is a suitablly chosen (simple) polynomial,

show how this problem can be transformed into one for Y[x] with homogeneous boundary conditions but where Y[x] satisfies an inhomogeneous equation.

have a go......
2. P must be such that Y(a) = Y(b) = 0. So we need P(a) = alpha and P(b) = beta. Those conditions are satisfied if

P(x) = alpha + (beta - alpha)(x - a)/(b - a)

--

Substituting y(x) = Y(x) + alpha + (beta - alpha)(x - a)/(b - a) into

y''(x) + p(x)y'(x) + q(x)y(x) = 0

gives

Y''(x) + p(x)[Y'(x) + (beta - alpha)/(b - a)] + q(x)[Y(x) + alpha + (beta - alpha)(x - a)/(b - a)] = 0

Y''(x) + p(x)Y'(x) + q(x)Y(x) = -[(beta - alpha)/(b - a)][p(x) + q(x)(x - a)] - alpha q(x)
3. cheers m8

really weird thing, i havn't used these forums in ages...let alone give out much rep. but you were the last person i repped so i got to rep a random person first....bla bla.

n e way thx
4. what is rep?

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Updated: December 12, 2005
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