Integrating seperable equation

[astonmartin]

Homework Statement

Separable equations

dy/dx = y * e^(sinx +cosy)

and

dy/dx = sin(x^y)

The Attempt at a Solution

For the first problem, I did dy/dx = y * e^(sinx) * e^(cosy) and separated. However, I can't figure out how to integrate e^(sinx)dx on the right. Did I do somehting wrong?

I have no idea what to do on the second one
Thanks for the help!

 

[astonmartin]

Help pleeeease

 

[Gib Z]

For the first one, my first impression would have been to let u= sin x, giving $$\int \frac{e^u}{\sqrt{1-u^2}} du$$ and then tried an integration by parts. Checking with the Integrator online however, it seems there is no elementary antiderivative for that function, so don't be surprised if the Integration by parts doesn't work out. Still try it though, because the Integrator has been wrong before.

The Second one isn't actually separable.

 

[HallsofIvy]

The second, y'= sin(x^y), is NOT separable.

The first is separable but gives integrals that cannot be integrated as elementary functions.
$$\int \frac{dy}{ye^{cos(y)}}= \int e^{sin(x)} dx$$
is the best you can do.

Thanks, Gib Z.

 

[Gib Z]

Small correction - The exponential term on the LHS should be in the denominator. =]