Solving Sin(x) Equation with ln(y), exp(y), and a Constant k

[Chatt]

Ive got the equation: sin(x) = ln(y) + (y^2)/2 + k

The k is a constant from an earlier integration. How do I isolate y? What makes it hard for me is that if i want to get rid of ln() i need to use exp() but then the other y is in exp() and if i want to get rid of that, the first y is in ln() again.

Sorry, for the bad english =) I hope you can help.

 

[Mentallic]

I'm afraid it's not possible to isolate y algebraically.

Taking the exponential of both sides: $$y=e^{sinx-\frac{y^2}{2}-k}=\frac{e^{sinx-k}}{e^{\frac{y^2}{2}}}$$

So now you have: $$ye^{\frac{y^2}{2}}=e^{sinx-k}$$

There is some method to solving for x in: $xe^x=y$
but the name of it has slipped my mind and whether it can be adapted to solve for y in your problem I'm unsure of as well. Hopefully someone else can help you with this.

 

[HallsofIvy]

Generally speaking, when you have a variable both "inside" and "outside" a transcendental function, there is no algebraic way to isolate that variable.

The "method of solving $xe^x= y$" is "Lambert's W function" which is defined as the inverse function to $xe^x$. That is, $W(xe^x)= x$ so $W(xe^x)= W(y)$ and $x= W(y)$.