# Also O.D.E stuck on integration step

1. Aug 27, 2005

### asdf1

In the problem
y` (cosh^2)x - (sin^2)y = 0, y(0)=pi/2
i'm also stuck on the integration step...
my first step is
dy/((sin^2)y)=dx/(cosh^2)x
however, i don't how to integrate the right side~

Last edited: Aug 27, 2005
2. Aug 27, 2005

### TD

You could try using the definition:

$$\cosh \left( x \right) \equiv \frac{{e^x + e^{ - x} }}{2}$$

3. Aug 27, 2005

### asdf1

ok, i'll try that~
thanks!

4. Aug 27, 2005

### lurflurf

5. Aug 29, 2005

### asdf1

wow~ i thought that only cos and sin have that kinda identity...
thanks! :)

6. Sep 3, 2005

### asdf1

hmm...
if i use $$\cosh \left( x \right) \equiv \frac{{e^x + e^{ - x} }}{2}$$ to integrate cosh^2x=(e^2x + e^(-2x) +1)/2,
then [integrated(cosh)dx]=[(1/4)e^2x+(1/4)e^(-2x) + x/2]
but does that equal tanhx?

7. Sep 3, 2005

### Fermat

I'm afraid not. It's the integral of sech²x that is equal to tanhx.

let y = tanhx
then
y = sinhx/coshx
y' = {coshx.(sinhx)' - sinhx.(coshx)'} / cosh²x
y' = {cosh²x - sinh²x} / cosh²x
y' = 1 / cosh²x
y' = sech²x

8. Sep 4, 2005

ok, thanks!