How can you solve for y in a difficult separable equation?

[Jamin2112]

Homework Statement

dy/dx = (y cos x) / (1+y²)

Homework Equations

Meh

The Attempt at a Solution

I've made it to this point:

ln(y) + y²/2 = sinx + C.

But we can't figure out to solve for y. It seems impossible with the ln(y) hanging around.

 

[PAllen]

As far as the differential equation goes, you've solved it. Does your problem statement requre you do express the solution as y=f(x) ?

 

[Char. Limit]

Theoretically, you can solve for y using the Lambert W function, defined as the inverse function to x*e^(x). But I doubt they'd make you do that.

 

[Jamin2112]

Theoretically, you can solve for y using the Lambert W function, defined as the inverse function to x*e^(x). But I doubt they'd make you do that.

Is their another way, brah?

 

[Jamin2112]

As far as the differential equation goes, you've solved it. Does your problem statement requre you do express the solution as y=f(x) ?

PAllen = Paul Allen?

 

[PAllen]

pallen = paul allen?

myob

 

[Char. Limit]

Is their another way, brah?

Not really. I checked wolframalpha, and that's the only way.

$$log(y^2)+y^2 = 2 sin(x) + 2C$$

$$e^{log(y^2)+y^2} = e^{2 sin(x) + 2C}$$

$$y^2 e^{y^2} = e^{2 sin(x)+ 2C}$$

$$y^2 = W\left( e^{2 sin(x)+2C}\right)$$

$$y= \pm \sqrt{W\left( e^{2 sin(x)+2C}\right)}$$