Solve for power series satisfying certian relations among each other

Question

Write $y_i = f_i(z) = \sum_{k=1}^{\infty} c_k^{(i)} z^k$ for $i=1,\ldots, i$ for four (formal) power series. The following relations are given between them \begin{align*} y_1 & = y_2 + y_3 \\ y_2 & = y_4^2 z^2 + 2y_4 z^2 + z^2 \\ y_3 & = y_4^2 z^2 + 2y_4 z^2 + z^2 \\ y_4 & = (3y_4 + 3)z. \end{align*} What is the solution, i.e. what power series (for what coefficients) fulfill these relations (and in general how to tackle such problems?)

Answer 1

In this particular system, one can start by solving the fourth equation for $z$ to get the rational function

$y_4(z) = \frac{1}{1-3z} - 1 = \frac{3z}{1-3z} \, ,$

whose Taylorseries at the origin is the unique formal power series to solve the fourth equation. The second and third equations show $y_2$ and $y_3$ to be equal, so $y_1 = 2 \cdot y_2 = 2 \cdot y_3$. If you insert $y_4$ above into the second or third equation, you can determine $y_2 = y_3$ as rational function, whose Taylorseries at the origin then gives you the rest of the (unique) solution. Here you may wish to use the binomial theorem to get

$y_2 = y_3 = \frac{z^2}{(1-3z)^2} \, . $

You can always try to insert the formal series into your equation and compare coefficients, but I cant tell you, when you will succeed that way.