For example, I have a pde like this: dx/dt = k dx/dy (the 'd' refers to the partial derivative). Now, if the first independent variable, t is related to another variable, T, by the relation, dt = dT/m (here, 'd' is the total differential and m is a constant), can I write the original pde as: m. dx/dT = k dx/dy?Is it possible to change the independent variable in a partial differential equation?

You can definitely do that. In cases like this, you can kind of treat the derivatives like, they're fractions, even though that's not technically the case. You're actually using the chain rule, which states that dx/dT = dx/dt*dt/dT, then just rearranging some variables.

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