Sturm Liouville and Self Adjoint ODEs

[Ed Quanta]

So how do I show that when we have a linear second-order differential equation expressed in self adjoint form that the Wronskian W(y1,y2)= C/p(x)

W=y1y2'-y1'y2, and C is a constant, and p is the coefficient where Ly=d^2/dx^2(pu) - d/dx(p1u) +p2u ?

I know Ly1=0 and Ly2= 0 if that helps at all.

 

[ReyChiquito]

why don't you start by writting down the definition for self-adjointness?

then take $(Ly,u)$ and integrate by parts.

 

[M98Ranger]

To show that the Wronskian of two solutions of a self-adjoint linear second-order differential equation is a constant multiple of the coefficient p(x), we can use the Sturm-Liouville theorem. This theorem states that for a self-adjoint differential equation of the form Ly = d^2/dx^2(pu) - d/dx(p1u) +p2u, the Wronskian W(y1,y2) = C/p(x) where C is a constant, if and only if y1 and y2 are solutions to the differential equation and satisfy the boundary conditions y1(a) = y2(a) = 0.

In this case, since we are given that Ly1 = 0 and Ly2 = 0, we can use the Sturm-Liouville theorem to conclude that the Wronskian of y1 and y2 is a constant multiple of p(x). This is because the boundary conditions y1(a) = y2(a) = 0 imply that y1 and y2 are linearly independent solutions, and thus their Wronskian is non-zero. The Sturm-Liouville theorem then tells us that the Wronskian is a constant multiple of p(x), which is what we wanted to show.

To further understand this result, we can also consider the physical interpretation of the Wronskian in the context of the self-adjoint differential equation. The Wronskian represents the rate of change of the solutions y1 and y2 with respect to each other. In a self-adjoint differential equation, the coefficient p(x) represents the density of a physical system, and the solutions y1 and y2 represent different modes of vibration. So, the Wronskian being a constant multiple of p(x) means that the rate of change of these different modes of vibration is directly proportional to the density of the system, which is a physically intuitive result.

In summary, we can show that the Wronskian of two solutions of a self-adjoint linear second-order differential equation is a constant multiple of the coefficient p(x) by using the Sturm-Liouville theorem and the boundary conditions given in the problem. This result has a physical interpretation in terms of the density and modes of vibration of the system.