Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 1A (Differential Equations)

The characteristic polynomial of this second-order linear differential equation factors as $r^2-r-6=(r-3)(r+2)$. Its two distinct roots give the general solution $y=A{e}^{3x}+B{e}^{-2x}$. Boundedness as $x\to \infty$ forces $A=0$, and the initial condition gives $B=1$.

Collecting the shifted terms gives a second-order difference equation with coefficients $1-h/2$, $-(2+6h^2)$ and $1+h/2$. Substitute $y_n=r^n$ to obtain its characteristic equation of a linear recurrence:For real $h\ne 0,2$, the roots and general solution areIn the small-step regime, indeed for $0<h<2$, the polynomial is positive at $r=0$, negative at $r=1$ and positive for sufficiently large positive $r$. Thus $0<r_{-}<1<r_{+}$. The bounded solution of a second-order constant-coefficient recurrence discards the growing root, and $y_0=1$ fixes its remaining coefficient:The Taylor series establishes the displayed approximation to the root. Raising that approximation to the $n$th power is a leading approximation, rather than a uniform relative approximation for arbitrarily large $n$: its relative error is controlled when $n{h}^2$ is small. For completeness, the degenerate values give $y_n=A+Bn$ when $h=0$, and $y_n=C13^{-n}$ when $h=2$.

Divide the second-order difference equation by $h^2$ and set $x=nh$. The first quotient is a central finite difference approximation to $y^{\prime \prime }(x)$; the quotient $(y_{n+1}-y_{n-1})/(2h)$ approximates $y^{\prime }(x)$. Their Taylor series errors are $O(h^2)$ for a sufficiently smooth function. This is therefore a finite difference method for the second-order linear differential equation in part (a).

At a fixed $x\ge 0$, take $h\to 0$ with $n=x/h$ integral. Since $\log r_{-}=-2h+O(h^3)$, the exact bounded discrete solution satisfiesLikewise $(1-2h{)}^{x/h}\to e^{-2x}$. The decay selected by boundedness in the linear recurrence relation becomes exactly the decay selected by boundedness in the ordinary differential equation.