Solving a Differential Equation with Substitution and Integration

[adam640]

I have been asked to solve the differential equation:

cotan(x)$$\stackrel{dy}{dx}$$ = y^-1 e^y2

Using substitution on the right hand side and by integrating tan(x) on the left hand side the answer I have got is:

ln|sec(x)| = - 0.5 e^-y2 + C

Is the answer ok to leave in this form? It has been a while since I have covered this topic.
Thanks, any help appreciated.

 

[grey_earl]

You can do this explicitely. Since sec(x) = 1/cos(x), you have (freely renaming the constant C)

2 ln | C cos(x) | = exp(-y²)
y = +/- sqrt [- ln ( 2 ln | C cos(x) | ) ]

Don't know if that's simpler, though.

BTW: Why does everyone use sec(x) and cosec(x) when it's much easier to use 1/cos(x) and 1/sin(x)? Never understood that.