Green's function for third order ODE

[squenshl]

I am trying to find a Green's function for a third order ODE.

($\lambda$ - d³/dx³ - $\mu$ d/dx)u = f(x) with boundary conditions u(0) = u_xx(0) = u_x(L) = 0 (L is finite).

This has solution u(x) = c₁e^m₁x + c₂e^m₂x + c₃e^m₃x

where m₁, m₂ and m₃ are the roots of the characteristic equation from the ODE (they are not important at the moment).

My question is how do I find the coefficients c₁, c₂ and c₃?

I know that it is going to be a 3 term piecewise function (as it is third order).

Do I use the u(0) = u_xx(0) = 0 BC's from 0 to s and u(0) = u_x(L) = 0 from s to L?

I need help on this please.
Cheers.

 

[gtoguy97]

The solution can be found by using the method of undetermined coefficients. To do this, you need to first solve the homogeneous version of the equation, which has solution u(x) = c1em1x + c2em2x + c3em3x. To find the coefficients c1, c2, and c3, you then need to use the boundary conditions to form a linear system of equations. For example, for the boundary condition u(0) = 0, you would have c1 + c2 + c3 = 0. Similarly, for uxx(0) = 0, you would have m1^3c1 + m2^3c2 + m3^3c3 = 0. The third boundary condition, ux(L) = 0, can also be used in this way. Once you have three equations, you can solve for the three unknowns c1, c2, and c3.