Recent content by John1987

That I know, but in that case, I get a complex-valued function u(x,t), right? I don't think Euler's equation can now be used to write this as sines and cosines, because I have only one imaginary "root" to my characteristic equation r-\lambda=0.

By the way: a hint was given together with the question that stated that it is allowed to divide by X(x)*T(t).

Well, assuming that this equation is correct: 2 V \left( \frac{X'(t)}{X(t)} \right)' \left( \frac{T'(t)}{T(t)} \right)' = 0. We know that either \left( \frac{X'(t)}{X(t)} \right)' or \left( \frac{T'(t)}{T(t)} \right)' should be zero. I think you'll agree with me on that. Assuming that...

Look, I'm trying to find non trivial solutions here, which implies that neither X(x) nor T(t) may be zero, since I assume: u(x.t)=X(x)*T(t). Given this constraint, it is allowed to divide by them. So, in my opinion the equation 2 V \left( \frac{X'(t)}{X(t)} \right)' \left( \frac{T'(t)}{T(t)}...

My computation for r: r = -\frac{\lambda*V}{V^{2}-c^{2}}\pm\frac{1}{2}*\sqrt{\frac{4*\lambda^{2}*V^{2}}{(V^{2}-c^{2})^{2}}-\frac{4*\lambda^{2}*(V^{2}-c^{2})}{(V^{2}-c^{2})^{2}}} Simplifying gives me: r =...

Well, I know that, as derived earlier: T(t)=(constant)*e^{\lambda*t}, where \lambda is an arbitrary constant. For T' and T'' I get: T'(t)=\lambda*constant*e^{\lambda*t} and T''(t)=\lambda^{2}*constant*e^{\lambda*t} By substituting in eq.2, I get, after dividing the total equation by...

Does anyone know how to determine the constant \lambda in the above equation? Usually, this can be done immediately after applying separation of variables, like in the Laplace equation. But now this doesn't seem to work. :confused:

Simply use l'Hôpital's rule to compute this limit. This indeed gives a value of 1 as x tends to infinity.

OK, this is quite helpful, thanks. I know that V is nonzero from the information given in the question. Also, \left( \frac{X'(x)}{X(x)} \right)' cannot be zero since this implies that X(x) is given by an exponential function, which gives, after applying the boundary conditions, a trivial...

From equation 4, I can now deduce that \frac{T'(t)}{T(t)} should be constant. This is because from eq.4, it follows that: \frac{T'(t)}{T(t)}=\frac{1}{2*V}*\frac{(c^{2}-V^{2})*(X(x)*X'''(x)-X''(x)*X'(x))}{X*X''(x)-X'(x)*X'(x)} The left hand side depends only on t and the right hand side...

Well, I don't see how to to do that right now. Also I'm not completely sure which equation you mean. Is it the second equation, as I just numbered them?

Last week, the professor gave a difficult PDE to solve as a bonus exercise, describing the motion of a conveyor belt. From experience, he knew that only 5% of the students (applied physics) is able to solve this problem. I got stuck and I really hope to get some help on this forum. This is the...