3.9 How to replace \(y(x)\) by \(e^{s(x)}\) in an ode?

Given ode \(y^{'''}=x y(x)\) we want to replace \(y(x)\) by \(e^{s(x)}\)

ClearAll[y,x,s] 
eq = y'''[x] == x y[x] 
eq = eq /. y -> Function[{x}, Exp[s[x]]] 
Simplify@Thread[eq/Exp[s[x]], Equal]
 

Gives \(x=s^{'''}(x)+s'(x)^3+3 s'(x) s''(x)\)