For y'=x-y, the field is horizontal at (2,2). Does this make y=2 a constant solution on an interval around x=2?

• No. On y=2 the slope is x-2, which is not always zero.
• Yes. A single zero slope makes the entire line a solution.
• Yes. Every constant function solves an equation with y'.
• No. Horizontal field segments rule out all solution curves.

An Euler table for y'=x-y starts at (x0,y0)=(1,3) with h=0.25. What is x1?

For y'=x+2 and y(0)=1, one Euler step with h=1 gives 3. The exact value at x=1 is 3.5. How does the approximation compare?

• An overestimate: 3 exceeds 3.5
• An exact value: 3 equals 3.5
• An underestimate: 3 is below 3.5
• No comparison is possible with these values

For y prime = y minus x, what does the field segment at (2, 2) look like?

• Rising with slope 2
• Flat and horizontal
• Falling with slope minus 2

For a smooth problem, assume Euler's nonzero leading global error dominates at a sufficiently small step size. What usually happens to endpoint error when h is halved over the same interval?

• It remains about the same size
• It grows to about twice its size
• It shrinks to about half its size

You sketch the solution of y prime = x minus y through (0, 1). How does it leave the point?

• Falling with slope minus 1
• Rising with slope 1
• Leaving flat with slope 0

When the exact solution's slope changes within an Euler step, why can the approximation drift from the solution?

• It uses the final slope for every point inside the step
• It holds the starting slope fixed during the whole step
• It replaces every changing slope with the current y-coordinate

For y'=x+1 with h=1, an Euler table has reached (x1,y1)=(1,1). What is the next approximation y2?

Kim starts at (0,0) for y'=x+1 with h=1. Kim uses the slope at x=1 and calculates y1=2. Which change gives an explicit Euler step?

• Start the table at x=1 while keeping its original starting y-value
• Set the step size equal to the slope at the next input
• Use the slope at the starting point before updating

Euler's method follows the exact solution for every differential equation and every step size.