Hello everyone, I have an exam in Linear differential equations and transforms soon and could do with some basic-ish help on how to solve second order diff equations.
The question states:
The function u(x,t) satisfies the equation
(d2U/dU2) = (1/c^2) (d2u/dt2) + (2b/c) (du/dt)
For 0 < x < l , t > 0 , and c > 0 , 0 < b < pi/l are constants.
Suppose u(x,t) is subject to homogeneous boundary conditions u(0,t) = u(l,t) = 0 and initial condition u(x,0) = 0 for 0 < x < l.
Use separation of variables to derive general solution:
u(x,t) = SUM(n-inf) [ Bn * exp(-bct) * sin(wn)t * sin((n pi x)/l)
Where Bn is the nth value of the constant B
And wn = c * sqrt ( (n^2 pi^2)/l^2 - b^2)
I so far have separated variables to find
X''T = 1/c^2 XT'' + 2b/c XT'
So
X''/X = 1/c^2 T''/T + 2b/c T'/T = -k^2
And
X'' + k^2 X = 0 => X(x) = Acoskx + Bsinkx
And similarly
T''+2bcT' + c^2*k^2*T = 0
Could anybody confirm that this is the right approach? And where do I go from here, how do I solve T?
Thanks very much, its much appreciated, and I apologise for the crude writing of equations I tried my best to make them look clear!