# Diffusion Equations : Separation of Variables

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4. (Separation) Seeking a solution u(x, t) = X(x)T(t) for the given PDE, carry out steps analogous to equations (3)?(6), and derive ODE's analogous to (7a,b). Take the separation constant to be ?K2, as we do in (6). Obtain general solutions of those ODE's (distinguishing any special ic values, as necessary) and use superposition to obtain a solution analogous to the solution (13) of (1 a). if the PDE cannot be separated, state that.

(b) uxx + 2ux = Ut HINT: In this case you should find that the value of K that needs to be distinguished Las we distinguished the case K. = 0 in (9) and (10)1 is N. = 1, not K = 0.

(Please view the attachment starting with hw5-1, hw5-1', and hw5-1''.)

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Separation of vairbles and diffusion equations are investigated. The solution is detailed and well presented.

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The equation is separable.

We start by defining:

Since X(x) is a function of x ...

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