Calculate $u(0.8)$ and $u(1.6)$ using the Forward Euler algorithm with time step $\Delta t = 0.8$. Plot the solution (i.e., a graph of $u$ versus time with line segments between the three points).
$u(0.8) \approx $

$u(1.6) \approx $

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values of solution:

##### Hint

Calculate your answers to at least 4 significant digits. This means that you should keep your intermediate answers to 6 or more significant digits to be safe.

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There should be something wrong with the plot you just drew. To find the problems, sketch the solution using a graphical approach. To do so, you'll need to calculate the equilibria, their stability, and determine how the speed of $u$ evolves with time.
Equilibria:

(Separate multiple answers by commas.)

Stability of equilibria:

(Enter `stable` for stable and `unstable` for unstable. Separate multiple answers by commas and enter in same order as equilibria.)

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Final values of curves:

Initial conditions of curves:

##### Hint

In this case, we are concerned only with the solution that has the initial condition $u(0)=0.5$, so you should draw just the one curve.

Online, drag $n_c$ to 1 to specify one curve, draw the point on the vertical axis, click the curve, if needed to change its shape (its speed profile), and drag the second point to specify the ending value of the solution curve. In this case, we are not concerned with the shape of the curve only that it starts and ends at the correct values.

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The problem with the Forward Euler solution is that we took too large of a time step. When $\Delta t = 0.8$, we keep going along the same slope for too long. We end up sailing past an equilibrium. Then, since we crossed the equilibrium, the next estimate of our direction is in the opposite direction, and it looks like the solution turns around. To avoid this problem, we must decrease the time step.

Redo the Forward Euler approximation, but this time halve the time step to $\Delta t=0.4$. Now, take four steps of the Forward Euler algorithm.

$u(0.4) \approx$

$u(0.8) \approx$

$u(1.2) \approx$

$u(1.6) \approx$

Plot your solution.

**Feedback from applet**

values of solution:

This time, the solution shouldn't cross the solution or turn around. (Well, OK, maybe it did a tiny bit. We'd have to make the time step even a little smaller to eliminate this error. But, you are probably glad we didn't take 16 steps with $\Delta t=0.1$. If you are bored some day, you could try it. At least the problems are not obvious from the graph.)

The important take-home message is that the Forward Euler algorithm could give you bogus results if you take time steps $\Delta t$ that are too big. You need to use ~~common sense~~ special mathematical know-how to check if Forward Euler (or any other computer program) is lying to you.

##### Hint

Your answers should be correct to within 4 significant digits, so be sure to keep 6 or more digits on all your intermediate answers. (It's OK to enter more than 4 significant digits in your answers as long as your answer is correct to 4 significant digits.)

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