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?
Best Answer
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 $$ \dfrac{dx}{ds} = 4 \\ \dfrac{dt}{ds} = 1 $$