Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 1 / b / ii

Set $z=x^3/3$ and write $y(x)=Y(z)$. Thenso the two first-derivative contributions cancel exactly, leavingThe Dirichlet spectrum of $-d^2/d{z}^2$ on this interval is $(3n\pi {)}^2$, $n\ge 1$. If $\lambda +a\le 0$, the linear or hyperbolic solutions cannot satisfy both zero endpoints nontrivially, so no additional modes have been omitted. ThusTo verify normalization, use $dz=x^{2}dx$:This proves weighted orthonormality and also transfers completeness from the ordinary Fourier sine series. The construction is the cubic-coordinate reduction of a singular Sturm-Liouville operator. The positive constant $a$ shifts the entire spectrum down; a zero eigenvalue occurs if $a=9n^2{\pi }^2$, so the no-zero-eigenvalue proviso in part (a)(iii) must be imposed separately if that inverse formula is used here.