Hey all, not sure if anyone will see this in time, but I've been stuck on a problem in a past paper for a while which I hope one of you might be able to help me with:
Considering a function U(x, t), use the method of separation of variables to find a general solution for U.
(dU/dt) = (dU/dx) + aU
Under boundry conditions U(x, 0) = Acos(kx).
The thing which gets me stuck is the constant in the RHS. I can solve the equation without it, but with it it becomes a nightmare. I have a similar problem with second partial derivates, although I'm guessing that the solution will be similar.
My attempt: Separating variables
U(x, t) = X(x)T(t)
(1/T)(dT/dt) = (1/X)(dX/dx) + a = Constant (K)
Solving LHS:
K = (1/T)(dT/dt)
ln(T) = Kt
T = A exp[Kt]
Solving RHS:
K - a = (1/X)(dX/dx)
ln[X] = x(K - a)
x = B exp[x(K - a)]
Which supposedly gives a general solution of the forum:
U(x, t) = C exp[Kt] exp[x(K - a)] = C exp[Kt + Kx - ax]
Which doesn't make much sense, and leaves me completely befuddled under the BC: U(x, 0) = A cos[Kx]
So, can anyone see where I'm going wrong?
Considering a function U(x, t), use the method of separation of variables to find a general solution for U.
(dU/dt) = (dU/dx) + aU
Under boundry conditions U(x, 0) = Acos(kx).
The thing which gets me stuck is the constant in the RHS. I can solve the equation without it, but with it it becomes a nightmare. I have a similar problem with second partial derivates, although I'm guessing that the solution will be similar.
My attempt: Separating variables
U(x, t) = X(x)T(t)
(1/T)(dT/dt) = (1/X)(dX/dx) + a = Constant (K)
Solving LHS:
K = (1/T)(dT/dt)
ln(T) = Kt
T = A exp[Kt]
Solving RHS:
K - a = (1/X)(dX/dx)
ln[X] = x(K - a)
x = B exp[x(K - a)]
Which supposedly gives a general solution of the forum:
U(x, t) = C exp[Kt] exp[x(K - a)] = C exp[Kt + Kx - ax]
Which doesn't make much sense, and leaves me completely befuddled under the BC: U(x, 0) = A cos[Kx]
So, can anyone see where I'm going wrong?