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
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:-
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
Below is the graph of the function y(x) which is a solution to the differential equation...
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