(Solved): B2. The non-zero function \( y(x) \) satisfies the Sturm Liouville problem \[ ...

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B2. The non-zero function \( y(x) \) satisfies the Sturm Liouville problem \[ y^{\prime \prime}+2 y^{\prime}+(1+\lambda) y=0 \] over the interval \( 0 \leq x \leq \pi \) with boundary conditions \( y(0)=y(\pi)=0 \), where a prime denotes differentiation with respect to \( x \) and \( \lambda \) is the eigenvalue. (a) (9 marks) Find the eigenvalues \( \lambda_{n} \) and the corresponding eigenfunctions \( y_{n}(x) \), and hence write down the general solution. (b) (9 marks) Determine suitable functions \( p(x), q(x) \) and \( r(x) \) such that the differential equation can be written in the standard self-adjoint form \[ \frac{d}{d x}\left[p(x) y^{\prime}(x)\right]+[q(x)+\lambda r(x)] y=0 \] and hence write down the differential equation in standard form. (c) (7 marks) Hence write down the orthogonality relation satisfied by the explicit eigenfunctions \( y_{n}(x) \) and \( y_{m}(x) \). Furthermore, calculate the value of the corresponding orthogonality relation satisfied by \( y_{n}(x) \) with itself (i.e., when \( n=m \) ).