A series solution of the differential equation: $\frac{d^2u(x)}{dx^2}+ u(x)^n = 0.$

Question

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.

Answer 1

The answer is relatively straightforward for a standard power series of the form, $u(x) = \sum a_l x^l$. The key is to use Mathematica's built-in series support. First, define $u$

u = With[{lmax = ...}, Sum[ a[l] x^l, {l, 0, lmax}] + O[x]^(lmax + 1) ]

where you set nmax as high as you need to recognize the pattern. For $n = 1$ (using you definition above), the differential equation

D[u, {x, 2}] + u

with lmax = 7 gives

(a[0] + 2 a[2]) + (a[1] + 6 a[3]) x + (a[2] + 12 a[4]) x^2 + (a[3] + 
20 a[5]) x^3 + (a[4] + 30 a[6]) x^4 +(a[5] + 42 a[7]) x^5 + O[x]^6

Note, Mathematica automatically collected the coefficients because we used O[x]^(lmax + 1), and correctly calculated the order of the neglected terms.