The number of initial conditions of Cauchy problem for PDE

This number, is it always equal to the order of the differential equation?

in which case, one can reduce this number?

If you are given $$\frac{{{d^n}y}}{{d{x^n}}} = f(x,y,y',y'', \ldots ,{y^{(n - 1)}})$$ then you need the values of $$y({x_0}),y'({x_0}),y''({x_0}), \ldots ,{y^{(n - 1)}}({x_0})$$ and the function itself. So you are right -- the number of initial conditions corresponds to the order of the differential equation. But this differential equation can, of course, be reduced to $$\frac{{dy}}{{dx}} = {y_1},\frac{{d{y_1}}}{{dx}} = {y_2}, \ldots ,\frac{{d{y_{n - 2}}}}{{dx}} = {y_{n - 1}},\frac{{d{y_{n - 1}}}}{{dx}} = f(x,{y_1},{y_2}, \ldots ,{y_{n - 1}})$$ using substitutions $${y_1} = y',{y_2} = y'', \ldots ,{y_{n - 1}} = {y^{(n - 1)}}.$$
Yes, we need $n$ conditions. –  glebovg Oct 20 '12 at 20:35