Are both answers for $xu_x + yu_y = 0$ valid?

Question

Solving this problem by the method of characteristic curves we have to solve the ODE

$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y}{x}$$

which gives us $$C = \ln(y/x)$$

where $C$ is constant.

Therefore, the solution should be

$$u(x,y) = f(\ln(y/x))$$

We can go further by noting that $e^C$ is also a constant which we can name $C_2$ giving us

$$C_2 = y/x$$

and the solution would give

$$u(x,y) = f(y/x)$$

Are both solutions valid?

Answer 1

They are the same solution but not the same function to express it. Using the same name for both functions is confusing. Call the second $g$, so $f(\ln(x/y))=g(x/y)$ For any chosen $f$ we have determined $g$ as $g=f\circ \ln$

Answer 2

I like to think of it that when:

$$C = \ln \left ( \frac{y}{x} \right ) \iff \frac{y}{x} = e^{C} = C_1$$

So you can just think of it as unwrapping the independent variables until you can't go any further.