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Consider the differential equation

$\frac{d^2u(x)}{dx^2}+ u(x)^n = 0.$

Let the solution be

$u(x) = u_0(x) + p u_1(x) + p^2u_2(x) + \cdots +p^m u_m(x).$

Now we are interested in substituting the above solution into the original differential equation and collecting the coefficients of $p$. Here, we may assume that $m$ and $n$ are positive integers.

How can we program this with a computer algebra system such as Maple/Mathematica?

Thank you.

  • 1
    What are the $u_i(x)$ and $p$ intended to be? Usually in solving a DE the series way, one uses a [power series (Frobenius)](http://mathworld.wolfram.com/FrobeniusMethod.html) or a [Chebyshev series (Clenshaw)](http://comjnl.oxfordjournals.org/content/6/1/88.short)...2011-04-28
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    You can solve this differential equation exactly: Multiply by $u'(x)$.2011-04-28
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    @Christian: yeah, for $n=1$ you have a trigonometric solution, and for $n=2$ and $n=3$, you have elliptic function solutions. Not sure about other values though.2011-04-28

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