Let z = z (x,y) be a function with x = x(s), y = y(t) satisfying the partial differential equation
(Ill write ddz/ddt for the partial derivative of z wrt t and
dz/dt for the total derivative of z wrt t, as I have no idea how to use Latex.)
ddz/ddt + 1/2s^2(dd^2z/dds^2) + s(ddz/dds) - z = 0
Using chain rule, show that z satisfies
-2z + 2(dy/dt)(ddz/ddy) + s((2dx/ds + s(d^2x/ds^2))ddz/ddx + s(dx/ds)^2 dd^2z/ddx^2) = 0
The Attempt at a Solution
So I wrote out my little "tree" with the variables and found
ddz/ddt = ddz/ddy * dy/dt
ddz/dds = ddz/ddx * dx/ds
Where I think im struggling is
dd^2/dds^2 = dd/dds[ddz/ddx*dx/ds]
Im not too sure how I should be computing this derivative and all attempts have given errors so far.
Im pretty sure all I need to do is work this derivative out then sub everything back into my original equation, tidy it up a little and then I get my desired result.
Anyone steer me in the right direction please.