partial differential equations – Solve a first order PDE using the method of characteristics ($ exp(u)u_x + frac{y}{x}u_y = 1$)

I’m solving the PDE:

$$exp(u)u_x + frac{y}{x}u_y = 1$$

With characteristics given by

$$frac{dx}{exp(u)} = frac{dy}{y/x} = frac{du}{1} $$

Giving a corresponding solution of

$$c1 = x^2 -2y^2exp(u)\ c2 = u-frac{y^2}{2x}$$

Thus

$$u(x,t) = frac{y^2}{2x}+ F(x^2 – 2yexp(u))$$ .

Is this right?