Eliminate a parameterization from system of equations

Question

Let $x(t) = x_0 e^{\lambda_1t} $, $y(t) = y_0 e^{\lambda_2t} $

A book I am reading has performed the following change to remove the parameter $t$:

Let $y = cx^{\lambda_2 / \lambda_1}$ where $c = \frac {y_0}{(x_0)^{\lambda_2 / \lambda_1}} $

I'm having trouble seeing exactly how to make this transformation. Can anyone point this out?

Answer 1

For $x_0 \ne 0$, $y_0 \ne 0$ and $\lambda_i \ne 0$ we can solve for $t$ and equate there: $$ \begin{align} t = \ln(x/x_0) (1/\lambda_1) &= \ln(y/y_0) (1/\lambda_2) \Rightarrow \\ y &= y_0 \, e^{\ln(x/x_0) (\lambda_2 / \lambda_1)} \\ &= y_0 \, (x / x_0)^{(\lambda_2 / \lambda_1)} \\ &= \frac{y_0}{x_0^{(\lambda_2 / \lambda_1)}} x^{(\lambda_2 / \lambda_1)} \\ \end{align} $$