This is a PDE. A partial differential equation. ODE45 CANNOT solve it, as it is not designed to solve that class of problem. (Note the fragment ODE in ODE45. It is there for a reason.) By the way, the error you got was completely unrelated to the problem with trying to use ODE45. That was just invalid MATLAB syntax, so your code failed long before ODE45 figured out that your problem was not something it is designed to solve. Similarly, dsolve cannot solve that class of differential equation.
So you are NOT on the right track. You need to use a tool designed to solve a parabolic PDE. What you have written is a fairly classical form. You might look at the tool PDEPE. It is designed to solve a PDE of that general family.
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