Nonlinear Systems & Linearization · Superchallenge problem

The system $x' = x + y^2,\; y' = -y + x^2$ has a saddle at the origin. Near the origin its unstable manifold is a curve $y = h(x) = a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + \cdots$ tangent to the $x$-axis. Find $a_5$.