I have this ODE:
d^2t/dx^2 = -S/K where S and K are constants. Boundary conditions are: T=T1 at x=0 T=T2 at x=L.
It looks so simple, but i'm having some issues. I used the auxiliary equation first to get the complimentary part: m^2=0, real roots at 0,0 means an equation of form: Tc=A+Bx with A,B as unknown constants.
Next, i'm trying to get the particular solution using the method which I believe is MUC. This is where i'm a bit confused. I assumed a solution in the form of: Tp=-S/K (constant). But then my general solution becomes: T(x)=A+Bx-S/K... and I don't think I can have two constants (both A and S/K) in there since one would be absorbed.
In this case, would I assume Tp=-S/K*x^2 since a constant already exists? But if I do this, I wind up with:
T = A + Bx -S/K*x^2
T(0) = T1 = A
T(L) = T2 = T1 + B*L - S/K*L^2 -> B = (T2/L - T1/L + S/K*L)
T(x)=T1 + (T2/L - T1/L + S/K*L)*x - S/K*x^2 which i'm being told is wrong, and that each S/K term should be S/(2K). I can't see why though.