**Example:** Consider the autonomous differential equation with the initial condition

.

**1.**- Find
.

Hint: You may want to sketch the Slope Fields of this differential equation.

**2.**- Find the first five terms of the Euler Approximation when .
**3.**- Is there a contradiction between the results of 1 and 2 ?
If yes, explain what happened.

**Answer:**

**1.**- First, let us draw the slope fields of the differential equation.

Since , any solution to the differential equation is increasing. This differential equation has one critical solution*y*=0. Since the initial condition satisfied by the solution to the IVP is*y*(0) = -0.1 < 0, then we have*y*(*t*) <0 for all*t*. We deduce from this that **2.**- Recall the formula which gives the euler's approximations to the solution
This gives the first five terms as:

**3.**- According to the result in Part 1, the solution to the given IVP should always be negative and according to the above Euler's approximation the first term is positive. This is our contradiction. As a matter of fact, according to the slope field, the Euler's approximation should continue to rise and even tend to , which even further contradicts the conclusion of 1.

The reason behind this is that is a large step. So, after the first shot, it shoots above the critical solution

*y*=0. Try to do the same problem with different step-sizes.

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