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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The equation is separable.
We start by defining:
Since X(x) is a function of x ...
Separation of vairbles and diffusion equations are investigated. The solution is detailed and well presented.
Heat Equation : Temperature Distribution on a Brass Rod
9. The temperature distribution u(x, t) in a 2-m long brass rod is governed by the problem
(a) Determine the solution for u(x, t).
(b) Compute the temperature at the midpoint of the rod at the end of 1 hour.
(c) Compute the time it will take for the temperature at that point to diminish to 5° C.
(d) Compute the time it will take for the temperature at that point to diminish to 1°C.
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