Derivation via Tangent Line  Alternative Derivations of Euler's  Derivation via Taylor's Theorem

Derivation via Numerical Integration

Recall that the general solution of the differential equation, with the assumption f(t,y(t)) = f(t) in equation (5.1), was presented in equation (5.3). By similar analysis, the more general form of the solution to equation (5.1) takes the integral form:

    Using this transformed problem, the focus turns from that of a differential equation to a problem of integration. Note that the (unknown) solution y(t) appears on both sides of the above equation, which can complicate things a bit.

    With the focus now on an integration problem, it makes sense to think about how one would evaluate the above integral numerically. Recall that the integral of a function is simply the area under the curve defined by the function. A very simple approximation of the area under the curve is just a rectangular approximation, where the width of the rectangle is taken to be the interval of integration and the height is taken to be the height of the function at the left-hand endpoint of the interval.

   Figure 5.1:   The integral of a function represents the area bounded bounded by the curve and the t-axis, (a). An approximation to the area is the rectangular region bounded by the axis and the height at the left-hand end of the interval, (b).

Figure 5.1 depicts the situation and the approach proposed. Namely, the true area under the curve is to be approximated by a rectangular region whose height is determined by the height of the function at the left-hand endpoint of the interval.

    Naturally, this intuitive approximation of the integral translates into a numerical scheme. For the interval over  , the equation for  becomes

Substituting in our approximation to the integral and using the assertion that  , this becomes

    This leads to our computed approximation

For the general sub-interval under consideration, the general form of this approximation is Euler's formula,

  
 Derivation via Tangent Line  Alternative Derivations of Euler's  Derivation via Taylor's Theorem
Wed Oct 28 11:42:13 EST 1998