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teroenza
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Homework Statement
By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter [itex]\lambda [/itex].
Homework Equations
The ODE is:
[tex] \frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}} [/tex]
I am told that the solutions have the form:
[tex] \phi = 1+ \lambda x + \sum_{n=3}^{12}a_{n} x^{3/2} [/tex]
The Attempt at a Solution
The way I have solved series equations before was to equate the coefficients of like powers of x. In this case, however, I have those x underneath the root, and I don't know how to go about doing this. I have the terms, via Mathematica. But I don't know how to get relationships between the coefficients.
Code:
p[x_] := 1 + \[Lambda]*x + Sum[Subscript[a, n]*x^(n/2), {n, 3, 12}];
pdd[x_] :=
Sum[Subscript[a, n]*(n^2 - 2*n)/4 x^((n - 4)/2), {n, 3, 12}];
LHS = pdd[x] - x^(-1/2)*(p[x])^(3/2)
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