Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 1

Seek an integrating factor $\mu$ such that the differential expression becomes $-(p{y}^{\prime }{)}^{\prime }$. Its leading coefficient requires $p=\mu (1-x^2)$, while the first-derivative coefficient requires $-p^{\prime }=3x\mu$. ThusChoose the positive solution $\mu =\sqrt{1-x^2}$. The Sturm-Liouville form is thereforewithThe weight is positive on the open interval. The endpoints are singular because $p$ vanishes there, but the prescribed boundedness of $y_n,y_n^{\prime }$ makes the integration-by-parts boundary terms vanish.

Multiply the Sturm-Liouville form for $y_n$ by $y_m$ and subtract the equation for $y_m$ multiplied by $y_n$. Integration givesThe boundary expression is zero because $p\to 0$ and the functions and first derivatives remain finite. Since $n,m$ are distinct positive integers, this provesFor the derivatives, apply integration by parts once to the same Sturm-Liouville form:Thus the new weight isThis is derivative orthogonality when the Sturm-Liouville potential vanishes. For $n=1$ the derivative is identically zero, which is consistent with the identity.

Put $Y(\theta )=y(\cos \theta )$, with $0<\theta <\pi$. The chain rule givesConsequently the original differential expression becomes $-Y^{\prime \prime }-2\cot \theta Y^{\prime }$. If $Y=F/\sin \theta$, direct differentiation givesTaking $F=\sin n\theta$ therefore yieldsThe apparent endpoint singularities are removable. Expansion at $\theta =0,\pi$ givesThus both endpoint regularity conditions hold. This is the Chebyshev polynomial of the second kind $U_{n-1}(x)$.

The trigonometric addition formula givesDividing by $\sin \theta$ and writing $x=\cos \theta$ gives the recurrence relationThe first two members are $y_1=1$ and $y_2=2x$. If $y_n$ has degree $n-1$ and leading coefficient $2^{n-1}$, then $2x{y}_n$ has degree $n$ and leading coefficient $2^n$, whereas $y_{n-1}$ has lower degree and cannot cancel that term. Mathematical induction provesFor example the sequence begins $1,2x,4x^2-1,8x^3-4x$, agreeing with the Chebyshev polynomial of the second kind.