Help with any of this is much appreciated. I tried multiple ways to figure out the problems and not once was I getting anywhere.
i. To find the general solution, I tried to use the characteristic equation but I don't think that is the correct way. ii. I believe I will be able to do this if I have the first problem done. iii. Have no idea how to even began this. And explanation would be awesome.
The adjoint form $((1+x^2)y')'=0$ is not particularly useful for solving the DE (it can be useful for determining weight functions for orthogonality relations among the various eigenfunctions). To solve your DE, expand it, giving $$(1+x^2)y'' + 2x y' = 0.$$ Substituting $u = y'$ gives $$(1+x^2)u' +2x u=0,$$ which is separable and has the general solution $u = \frac{c}{1+x^2}$. Integrating again gives $y = c_1\tan^{-1}x + c_2$.
For (ii), note that from the above, $y' = \frac{c_1}{1+x^2}$, so that the boundary conditions become $$a_1y(0) + a_2y'(0) = a_1c_2 + a_2c_1 = 0, \quad b_1y(1) + b_2y'(1) = b_1\left(\frac{\pi}{4}c_1+c_2\right) + b_2\left(\frac{c_1}{2}\right) = 0,$$ or \begin{align*} a_2c_1 + a_2c_2 &= 0,\\ \left(\frac{\pi}{4}b_1+\frac{b_2}{2}\right)c_1 + b_1c_2 &= 0. \end{align*} This is a linear system in $c_1$ and $c_2$. When does a linear system have a nontrivial solution?
