Partial differential equation

Question

u_x=(1/sqrt(-y))*u_y

what is the general solution for u?

Answer 1

We looking for general solution as product of 2 functions: u(x,y) = f(x) g(y). Then u'_x=f'(x)g(y) and u'_y = f(x)g'(y).

We now substitute in given equation: f'(x)g(y) = 1/sqrt(-y)f(x)g'(y). then we prezent as quotions:

f'(x)/f(x) = 1/sqrt(-y)g'(y)/g(y) = C (constant).

We get ODEs: f'(x)/f(x) = C; ∫df/f = ∫Cdx; lnf(x) = Cx + LnD, f(x) = De^Cx, D is arbitrary constant;

Then second ODE for g(y): 1/sqrt(-y)g'(y)/g(y) = C, g'(y)/g(y) = Csqrt(-y); ∫dg(y)/g(y) = ∫Csqrt(-y)dy;

lng(y) = - 2C/3sqrt(- y³) + lnE, g(y) = Ee^{- 2C/3sqrt(- y^3)}, where E is arbitrary consant.

At last u(x,y) = f(x)·g(y) = De^Cx·Ee^{- 2C/3sqrt(-y^3)},u(x,y) = Ae^{Cx - 2C/3sqrt(-y^3)}, wher A= D·E arbitrary constant