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How to find ratio of no. of degree 14 & 10 polynomial satisfying some differential equation

How to find ratio of no. of degree 14 & 10 polynomial satisfying some differential equation.
Any help will be helpful.

A:

Differential equations of the form $f» = 0$ have finite number of solutions. There are $10$ linearly independent degree $14$ solutions and $14$ of degree $10$. You can verify that when you put them together as a single sequence, you get a basis of the space of solutions and hence that the dimension of the space is $20$ if the equation is of order $6$, $30$ if it’s of order $7$, etc.
Let’s start with a general differential equation, of order $6$, of degree $4$:
$$f»» + gf» + hf = 0$$
that needs to be solved. There are $4$ linearly independent degree $14$ solutions, so we have $10$ solutions in total.
Let us write down the general solution to this equation in terms of $x$. We have
$$ f = \sum_{k=0}^4 a_k x^k \implies f = x^4 + a_0 x^3 + a_1 x^2 + a_2 x + a_3 + a_4 x^{ -1} + a_5 x^{ -2} + a_6 x^{ -3} + a_7 x^{ -4}$$
If we try to substitute $x = 0$, we get $a_0 = 0$: the $0$’th derivative of the left-hand side vanishes. We can also plug in $x = 1$ and calculate $a_3$:
$$ f = 1^4 + a_0 1^3 + a_1 1^2 + a_2 + a_4 + a_5 + a_6 1^{ -1} + a_7 1^{ -2} \implies f = 1^4 + a_0 1^3 + a_1 1^2 + a_2 + a_4 1^{ -1

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