Series solution to nonlinear differential equation

  • #1
teroenza
195
5

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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  • #2
teroenza said:

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)
Do you mean [tex] \phi = 1+ \lambda x + \sum_{n=2}^{12}a_{n} x^{n} [/tex]?
 
  • #3
No, but I did make an error in the above. It should be:

[tex] \phi = 1+ \lambda x + \sum_{n=3}^{12}a_{n} x^{n/2} [/tex]

But I can't figure out how to edit my original post.
 
  • #4
That suggests to me the substitution x = u2. Haven't tried it, though.
 
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