Converting the Answer of an Integral to the For of cosh x

Question

I've integrated the following and function and the given answer is in the form of A cosh((x+a)/A). How do I simplify my answer to get this answer?

Integral

y^2/(1+((dy/dx)^2)) = c

Answer I got:

y= a*e^tx + b*e^(-tx)

Where a and b are constant and t=((c-1)/c), c is the constant above

Answer 1

this is a non-linear first order differential equation,,,,,

is there some method, since "integration" is not possible

on the equation, that you used?

I do know that d(cosh(x))/dx=sinh(x) and 1+sinh²x=cosh²x

by observation and trying to make y(x) most flexible

let y(x)=Acosh((x+a)/A)

then y²(1+y'²)=c

is [A²cosh²((x+a)/A)]/[1+sinh²((x+a)/A)]=c

c=A²,,,,,c>0

A=+/-√c

Answer 2

When integrating, I got the following:

y= (1/2 )c e^(-((2 x)/√c) - 2 c1) e^(((2 x)/√c) + 2 c1) which is basically cosh( (x+a)/A), you can see what is a and A?!)

In your case instead of a and b you could take log(a) and log(b).

Hope this helps. If not let me know.