What is the trick to Taylor expand this function to 4th order?

Question

The function is $ u(x,y)= -x-y-xyu^3$, and I want to Taylor expand $u(x,y)$ around (0,0) in powers of x and y to 4th order.

To first order, I differentiated implicitly, and the expansion is:

$$u(x,y) \approx -x-y $$

How can I go up to 4th order? Is there a trick to notice here? The work is already a nightmare at the 2nd order expansion, and I don't think the point of the question would be to do it with brute force to 4th order.

The hint given in the question was:

Hint: Substitute a power series for u into the equation, and determine the coefficients.

Thanks,

Answer 1

It may help to write out $u$, at least partially.

$$ u(x,y) = a_{00} + a_{10}x + a_{01}y + a_{11}xy + \cdots $$ Plug this into the right hand side of your equation but keeping only terms up to the fourth order. You don't have to write out everything because that third power is already multiplied two powers (one for $x$ and one for $y$). In fact what I've written above should be enough. (Do you see why?) At that point you should be almost done.

Does this help?

Answer 2

Since the third term is greater than fifth order, the fourth order expansion is equal to the first order expansion.