systems of linear differential equations: how to plot solutions y(t) versus x(t)?

Question

In the picture below from the Shone book, I don't get how "t is eliminated". When you write both expression in terms of the common entity e^t, you don't get this result. In some literature they say you have to divide or calculate the derivative of y with respect to x via the implicit function theorem, but I don't get how that is related.

So, how is it done? And why does my method not work or is incorrect?

Thanks!

Answer 1

$x(t) = 2e^{2t} = 2 (e^t)^2 = 2 \big(\frac{y(t)}{3}\big)^2 \implies y^2(t) = \frac{9}{2} x(t)$

Answer 2

You should get this result when $t$ is eliminated because $$y=\sqrt{\frac{9x}2}=\sqrt{\frac{9(2e^{2t})}2}=\sqrt{9e^{2t}}=3e^t$$ You can get this result by solving for the common parameter $e^t$ $$e^t=\frac{y}3=\sqrt{\frac{x}2}$$