方向场与数值方法

Reading a direction field

Euler's method

Accuracy and refinement

Qualitative plus numerical

A direction field for y prime = f(x, y) draws what at each point of the plane?

In a direction field, an equilibrium solution appears as what?

Euler's method with step size h updates the approximate solution by which formula?

The global error of Euler's method over a fixed interval is proportional to which power of h?

An equilibrium in a direction field has nearby solution curves approaching it as x grows. What is this equilibrium called?

To approximate a solution from x = 0 to x = 2 with step size h = 0.25, how many Euler steps are needed? Answer with a number.

For y prime = y with y(0) = 1 and h = 1, Euler gives y_1 = 1 + 1 times 1 = 2. What does it give for y_2 at x = 2? Answer with a number.

In the field for y prime = x + y, the segment at the point (1, 2) has slope 3. As a number, what is it? Answer with a number.

Runge-Kutta methods improve on Euler by what device?

Comparing Euler approximations at h and h/2, the difference estimates what quantity? Answer with two words.