# Relating terms in differential equation with power series

Having problems with a task on a differential equation containing a power series.

Given

$$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$ $$\frac{dy}{dt} = \lambda y$$ $$x(y) = y + \sum_{n=2}^{\infty}a_n y^n$$

the goal is to find a expression for $a_n$ in terms of $b_n$ and $\lambda$.

I have solved $y = C_1 e^{\lambda t}$ and tried to find something useful from $$\lambda y + \sum_{n=2}^{\infty}\lambda n a_n y^n=\lambda x + \sum_{n=2}^\infty b_n x^n$$ but have so far failed.

It seems I forgot to add a "hint" given with the task. Too make sure we are on the right track we are supposed to find that $$a_3=\frac{2b_2^2+\lambda b_3}{2 \lambda^2}$$

-