Show that when Euler's method is used to approximate the...

Show that when Euler's method is used to approximate
the solution of the initial value problem
$$y ^ { \prime } = - \frac { 1 } { 2 } y , \quad y ( 0 ) = 3$$
at $$x = 2$$, then the approximation with step size h is
$$3 \left( 1 - \frac { h } { 2 } \right) ^ { 2 / h }$$.

Key Concepts

Recurrence Relation

In Euler's method, the approximation is constructed through a recurrence relation that updates the solution at each step. For a general differential equation y' = f(x, y), the recurrence formula is y??? = y? + h·f(x?, y?), which, when applied repeatedly, builds an estimate of the solution over the interval of interest.

Euler's Method

Euler's method is a numerical technique for solving ordinary differential equations by approximating the solution incrementally. It uses a tangent-line approximation at discrete points to advance the solution, updating the current value by adding the product of the step size and the derivative evaluated at that point.

Initial Value Problem

An initial value problem consists of a differential equation along with a given starting value for the unknown function. This setup guarantees the existence of a unique solution and defines the problem that numerical methods, like Euler’s method, are designed to approximate.

Step Size and Discretization

The step size, denoted by h, determines the spacing between successive points in the discrete approximation of the continuous domain. A smaller step size generally leads to a more accurate approximation because the linear assumptions made over each interval are more valid when the interval is narrow.

Exponential Approximation

For linear differential equations, the repeated application of Euler's update creates a product of factors that, in the limit of step size approaching zero, behaves like an exponential function. This connection illustrates how the discrete process mirrors the continuous exponential decay or growth described by the exact solution.

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