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1. Homework Statement

1. Homework Statement

It's to do with mirages, but I don't think the physics context is too important. It's also possible that the solution doesn't involve differential equations, and my method is completely wrong. I've been given that:

##A = \frac{n(1+ay)}{ sqrt(1+(y')^2)}##

where y' is dy/dx. I have to show that

##y = -\frac{1}{a} + \frac{A}{na}## cosh( ##\frac{na}{A}##(x-x

_{0}))

A, n and a are real constants.

## Homework Equations

## The Attempt at a Solution

I tried rearranging it to get dy/dx = ... , by multiplying by the denominator then dividing by A, squaring both sides, subtracting 1 and then square rooting. I ended up with:

##y'=\sqrt{\frac{n^2 (1+ay)^2}{A^2} -1}##

And that's separable. So

##dx = \frac{1}{\sqrt{\frac{n^2 (1+ay)^2}{A^2} -1}}dy##

And then if I integrate both sides by doing substitutions like u =1+ay, I get ln of something. Nowhere near the show that result.

Checked the rearrangement so many times. What did I do?