A differential equation: dy/dx = e^ysinx^x / ysecx

dy/dx = e^ysinx^x / ysecx

rearranging it we get y/e^y dx = sinx^2cosxdx

after taking the integral of each side we end up with

-(y+1)/e^y = 1/3sinx^3 + C

Not sure how to go about rearranging this one for y

1) Notation is very unclear. Make sure you write what you mean. For example, is sinx^2 supposed to be \(\displaystyle \sin(x^{2})\) or \(\displaystyle \sin^{2}(x)\) which is the same as \(\displaystyle [\sin(x)]^{2}\) or something else? When you write clearly, you often manage to unconfuse yourself. Another example, 1/3sin(x). Is that \(\displaystyle \frac{1}{3\sin(x)}\) or \(\displaystyle \frac{1}{3}\sin(x)\). Please write clearly.

2) Why do you think you should solve for "y"? It looks fine and you really cannot improve its appearance.

-(y+1)/e^y = (sinx)^3/3 + C

I always thought that when solving for a differential equation you were suppose to solve y in terms of x.

Hello, Clifford!

\(\displaystyle -\frac{y+1}{e^y} \:= \:\frac{1}{3}\sin^3\!x + C\) . . . . . Correct!

I always thought that when solving for a differential equation.
you were suppose to solve y in terms of x.

Click to expand...

We can try to solve for y, but usually it is not possible.