Differential Equations | Part IA, 2003

Consider the differential equation

dxdt+Kx=0\frac{d x}{d t}+K x=0

where KK is a positive constant. By using the approximate finite-difference formula

dxndt=xn+1xn12δt\frac{d x_{n}}{d t}=\frac{x_{n+1}-x_{n-1}}{2 \delta t}

where δt\delta t is a positive constant, and where xnx_{n} denotes the function x(t)x(t) evaluated at t=nδtt=n \delta t for integer nn, convert the differential equation to a difference equation for xnx_{n}.

Solve both the differential equation and the difference equation for general initial conditions. Identify those solutions of the difference equation that agree with solutions of the differential equation over a finite interval 0tT0 \leqslant t \leqslant T in the limit δt0\delta t \rightarrow 0, and demonstrate the agreement. Demonstrate that the remaining solutions of the difference equation cannot agree with the solution of the differential equation in the same limit.

[You may use the fact that, for bounded u,limϵ0(1+ϵu)1/ϵ=eu|u|, \quad \lim _{\epsilon \rightarrow 0}(1+\epsilon u)^{1 / \epsilon}=e^{u}.]

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