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Given the PDE $\frac{\partial\rho}{\partial t} + 4 \frac{\partial\rho}{\partial x}= 0 $ determine the characteristics.

I understand how to solve this PDE using the method of characteristics as $\rho=f(x-4t)$, but I do not understand what is meant by "determine the characteristics".

Which equations are the characteristics?

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If you're already solving the PDE with the method of characteristics, than you're computing the "characteristics"; the solution is a union of characteristic curves.

They are integral curves of a vector field which the solution is tangent to at every point. In other words, the solution is constant along them and the equation reduces to a ODE along the characteristic curves.

In your case, the curve $x(s), t(s)$ such that $$ \dfrac{dx}{ds} = 4 \\ \dfrac{dt}{ds} = 1 $$

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