Differential Equations Exercises

Answer

Since we're asked about error, we'll need to know the exact value of f (2). Thinking backwards, f (x) = 2x² + C. Since f (1) = 3,

2(1)² + C = 3

so C = 1 and f (x) = 2x² + 1. The exact value in question is

f (2) = 2(2)² + 1 = 9.

Euler's method with step size 0.5 gets us this table:

The error is 1, since the exact value of f (2) is 9 but Euler's method approximated f (2) ≈ 8.

Euler's method with step size 0.25 gets us this table:

With step size 0.25 Euler's method approximates f (2) ≈ 8.5. The error in this case is 0.5.

Answer

If f is concave up, the tangent line to f at a is below the graph of f. This is true whether the slope of the tangent line is negative, zero, or positive.

The moral of the previous exercise is that when f is concave up, a tangent line approximation is an underestimate. Since Euler's method is a bunch of tangent line approximations stuck together, Euler's method will also provide an underestimate, regardless of how many steps are used.

When f is concave down, a tangent line approximation is an overestimate. Since Euler's method is a sequence of tangent line approximations, Euler's method also provides an overestimate, regardless of how many steps are used.

Remember when we drew, on a slope field, solutions to a differential equation that passed through a particular point? If we can sketch a solution, we can tell whether it's concave up or concave down. That's all we need to tell whether Euler's method will overestimate or underestimate.

Answer

If we draw the solution and slope field, we see that this solution is concave up:

This means Euler's method will produce an underestimate to the value f (2).

Answer

This is a bit tricky. The solution is concave up which would make us think that Euler's method would produce an underestimate.

However, the solution to this equation is a line, so all of our approximations will be exact. This means Euler's method will actually give us an exact answer.

Answer

Draw the solution to this differential equation that passes through (0, 5):

This solution is concave up, so Euler's method will produce an underestimate to the value f (1).

Answer

This is a trick question. The solution to that passes through the point (1, 1) is the line y = x.

Since the tangent line to a line is that line, Euler's method in this case will produce the exact value of f (2).

Answer

The solution that passes through (5, -1) is y = ln x – ln(5) – 1 which is concave down.

Therefore Euler's method will produce an overestimate to f (5.9).