User talk:Gareth Owen/WKB approximation

That looks like good work Gareth. I would change the start a little to make it more general.

Something like.

The WKB approximation is a method for approximating 2nd order linear differential equations. The general such homogenous equation is

a(x) \frac{d^2 y(x)}{dx^2} + b(x)\frac{d y(x)}{dx} + c(x) = 0

if we write b' = exp\int b/a dx this becomes

a(x)/b'(x) \frac{d }{dx}(b'(x)\frac{d y(x)}{dx}) + c(x) = 0

writing c' = b^c/a this becomes

b'(x) \frac{d }{dx}(b'(x)\frac{d y(x)}{dx}) + c'(x) = 0

Now we change the independent variable to x'= int(1/b'(x)dx) so that \frac{d}{dx'}= b'(x)\frac{d}{dx}. The equation is now in standard form

\frac{d^2 y(x)}{dx^2} + c'(x') = 0