I have a problem with an experimental data analysis to obtain unknown parameters. The experimental data can be described by the equation: $y_k-y_{k-1}=\eta_k (x_k^{1-\alpha_k}-x_{k-1}^{1-\alpha_k})$ where $y_k$s and $x_k$s are known as they are experimental outputs and inputs, respectively. The data is bounded by $y(0)=0$ and $x(0)=0$.
Set $k=1$ to obtain the first non-zero data point.
$y_1=\eta_1 x_1^{1-\alpha_1}$, taking the logarithm of both sides produces $\log(y_1)=\log(\eta_1)+(1-\alpha_1)\log(x_1)$, which is a line equation in $\log y-\log x$ domain.
Then I can calculate $\alpha_1$ by curve-fitting a line between the first and second data points. However, in the subsequent terms, I got stuck.
Could someone help me with how can we calculate the sequence of $\alpha_k$'s?