Verifying that a function is a solution to a given differential equation

Question

I'm given the differential equation $y-(y'-1)e^{y'}=0$ and I'm told that $x=e^{t+1}, y=(t-1)e^{t}, $ are solutions . How do I verify that? I tried just taking the derivative of $y$ and then doing what I'm given here $y-(y'-1)e^{y'}=0$, and hopefully get an equality, but that didn't work. And also if that did worked why would I also be given what $x$ is (considering that I wouldn't use it in the equation)?

Answer 1

$x = e^{t+1}, y = (t-1)e^t$ is a single solution in parametric form, not two different "solutions".

$y'$ means $\frac{dy}{dx}$. And

$$ \frac{dy}{dx} = \frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)} $$

Use this to verify or reject the given solution.