Question

Below is the graph of the function y(I) which is a solution to the differential equation due = f(,y). The 2-values of the labeled points below are -3, -2, and 1.7 respectively. Suppose that for geometric reasons we also know the y-values of points A and B. I wish to use Euler's method to calculate the y-value of point C. The starting point for Euler's method can either be point A or point B. 1.5 А с 0.5 B 0 -0.5 (a) What advantages/disadvantages are there to picking A to be the starting point? (b) What advantages/disadvantages are there to picking B to be the starting point?

Below is the graph of the function y(x) which is a solution to the differential equation dy dx = f(x, y).

please help me, thanks so much

Answer 1

In Euler's method, if we observe graphically, what we are doing basically is extending the lines from one reference point to another by taking the initial point as obtained after the extension of previous line and the slope same as the slope of the tangent at that point.

For better understanding, see that diagram:-

у Euler approximation Wi A3 A2 Az A W 2 A1 w 1 Ao Exact function y=f(t) y. to t t t 1

A) Now, if we take A as the starting point, we can see that we have the advantage that distance between A and B is less i.e. 0.5 units so we could take much less step size, it saves our time too as tedious calculations got reduced and we achieve better approximation.
Disadvantage is that there is a maxima between the two points. Usually for Euler's method, path with only either increasing path or decreasing path is preferred to maintain least distance between the Euler's approximation line and the actual curve so that accuracy can be improved.

B) Now, if we take B as the starting point, the advantage we have is that the curve is having only a decreasing path, there is no maxima or minima in between So accuracy will be better
Disadvantage is that the distance between B and C is much more than that of A and C i.e.1.5 units so that would be lead to a larger step size and lead to large errors as error in approximation is directly proportional to the square of the step size or if we took smaller step size, then it would become too lengthy and time- consuming.

General advantages and disadvantages of Euler's methods are as follows:-

Advantages:
1. Euler’s Method is simple and direct
2. Can be used for nonlinear IVPs

Disadvantages:
1. It is less accurate and numerically unstable.
2. Approximation error is proportional to the step size h. Hence, good approximation is obtained with a very small h. This requires a large number of time discretization leading to a larger computation time.
3. Usually applicable to explicit differential equations