Constructing the Green's Function for Non-trivial Solutions

[evinda]

Hello! (Wave)

Can the Green's function be contructed in the case when the homogeneous problem has non-trivial solutions? Justify your answer.

Try to construct the Green's function for the following problem:$$y''+y= \cos x , y(0)=y(\pi)=0$$

The corresponding homogeneous problem has solutions that are given by the relation $y=C \sin x$. So if we take into consideration the boundary conditions, we cannot find two linearly independent solutions of the homogeneous problem.

Thus the Wrosnkian will be zero, and so we cannot apply the formula of the Green's function . The same would also hold for any problem, the homogeneous corresponding of which has non-trivial solutions.Is my justification correct? Or am I wrong? (Thinking)

 

[gtoguy97]

Yes, your justification is correct. The Green's function cannot be constructed when the homogeneous problem has non-trivial solutions because it relies on being able to find two linearly independent solutions of the homogeneous problem. Since we cannot do that in this case, the Green's function cannot be constructed.