Supposedly simple differential equation

[popo902]

Homework Statement

Find the solution of
dy/dt = 1/ (e^y -t), y(1) = 0

Homework Equations

The Attempt at a Solution

i tried separating the equation, but the subtraction gets in the way
well this is what i have
y = t - 1 + C/e^t
i solved for t then i put that into the single order ODE formula
but the answer is y = arccosht
...so...
can someone give me a hint on how to start this?

 

[HallsofIvy]

Yes, the fact that an equation is not separable does make it hard to separate!

Here's what I would try. Since it is the $e^y- t$ that is the problem, let $u= e^y- t$ so that
$$\frac{dy}{dt}= \frac{1}{u}$$
and since $e^y= u+ t$
$$e^y\frac{dy}{dt}= \frac{du}{dt}+ 1$$
$$(u+ t)\frac{1}{u}= \frac{du}{dt}+ 1$$
$$\frac{du}{dt}= \frac{u+t}{u}- 1= \frac{t}{u}$$
which is separable.

 

[dextercioby]

Well, you can switch the variables and try to find t(y). The ODE which results

$$\frac{dt(y)}{dy} + t(y) = e^y$$

should be very easy to integrate, right ?

 

[HallsofIvy]

Well, if you want to do it the easy way!