I have done a linear stability analysis for a system of coupled PDEs. The growth rate of perturbations, $\lambda$, satisfies an equation $f(\lambda)=0$. Now I want to find the leading order terms in the growth rate. I did two methods to find the growth rate as a function of the wavevector $q$.
First method: expanding $\lambda$ as $\lambda=\lambda_0+\lambda_1 q+\lambda_2 q^2+\lambda_3 q^3+\lambda_4 q^4$, and then subing the expansion in $f(\lambda)=0$ and solving $f(\lambda)=0$ for different powers of $q$.
Second method: Solving $f(\lambda)=0$ for $\lambda$, and then expanding $\lambda$ for different powers of $q$.
I expected the two methods to give the same result, but they did not! I wonder why? And which method is correct.
I am asking this in the physics section because I don't understand where the difference comes from and what each method mean.
Here is the equation I tried to solve for $\lambda$.
$\lambda + D q^2 + 2 a - \frac{r \: e \:m}{6 J} + \lambda \frac{(- r \: e \:m) q^2 (J - e)^2}{2(J^3 q^2 \lambda + 2 r \: e \:m)} =0$
in this equation $r<0$.
