Euler's Method: Answer to Example1

Example: Consider the autonomous differential equation with the initial condition

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1.
Find

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

2.
Find the first five terms of the Euler Approximation when tex2html_wrap_inline105 .
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 tex2html_wrap_inline107 , 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

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2.
Recall the formula which gives the euler's approximations to the solution

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This gives the first five terms as:

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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 tex2html_wrap_inline123 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 tex2html_wrap_inline125, which even further contradicts the conclusion of 1.

The reason behind this is that tex2html_wrap_inline105 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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