The discussion focuses on finding the eigenvalues for the differential equation $$y''+\lambda y=0$$ with boundary conditions $$y(0)=0$$ and $$y'(0)=\frac{y'(1)}{2}$$. The eigenvalues are determined to be complex, derived from the relation $$\cos{(\sqrt{\lambda})}=2$$, with the only real eigenvalue being $$\lambda=0$$. The participants clarify that the characteristic equation should be $$m= \pm \sqrt{-\lambda}$$, and they explore the implications of complex eigenvalues and the nature of the solutions.
PREREQUISITESMathematicians, physicists, and engineering students focusing on differential equations, particularly those interested in eigenvalue problems and their applications in various fields.
I like Serena said:That sounds very plausible. (Mmm)
Actually, $y=0$ is always a solution, regardless of the value of $\lambda$, so $\lambda=0$ is not really the corresponding eigenvalue.
So it makes sense to leave that one out.
Furthermore, the eigenfunctions for $-\ln^2(2\pm\sqrt 3)$ are identical.
That is because $$\frac 1{2-\sqrt 3} = 2+\sqrt 3$$.
So we might consider them as the same.
I like Serena said:Have you tried to calculate the Wronskian for 2 eigenfunctions of which 1 has an imaginary eigenvalue? (Wondering)
mathmari said:So should I try it for $u(x)=\sin{(x(-i \ln{(2+\sqrt{3})}+2 \pi n))}$ and $v(x)=v^*(x)=(2+\sqrt{3})^x-(2+\sqrt{3})^{-x}$ ?
I like Serena said:Yes... (Sweating)