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My professor gives us the transport equation with an initial condition and use the method of characteristics to solve. However, when he asks us to find the ODE satisfied by characteristic curves, he uses

$$X(t;x_0).$$

Why is the function with parameters $t$ and $x_0$? and Why is there a semicolon in the function definition ?

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The method of characteristics can be thought of as reducing transport equations, wave motion, to a family of ODE initial value problems, particle motions. Here $X(t;x_0)$ means "The $x$-position at time $t$ of the particle that started at $x = x_0$." The method of characteristics says that if you can determine the position of all of these particles, then you have solved the related PDE and can write down the equation of the wave. The $x_0$ argument is separated by a semicolon because it isn't used as an explicit argument of the function $X(\cdot;x_0)$, but merely as an indicator to remember where that specific trajectory started.

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