i ONLY need help with #4. How do I find errors? For the initial value problem y\

Question

i ONLY need help with #4. How do I find errors?

For the initial value problem y'+y=2ex ; y(0)=7

1. Solve analytically for the exact particular (solve for C!) solution (it is a linear equation).
2. Solve using Euler on the interval 0?x?2 with 50 data points and then with 200 data points.
3. Solve using Improved Euler on the interval 0?x?2 with 50 data points and then with 200 data points.
4. Use the exact solution you computed to find the error in each numerical approximation at x = 2. Divide the errors in the Euler method with the two different mesh sizes and show the error was reduced by a factor of about the expected amount (what is that amount and why?). Do the same for the Improved Euler.

Explanation / Answer

The error is going to be the difference between your approximation (eulers, and improved eulers) and the exact value, both evaluated at 2. When you evaluate this difference with the different mesh sizes, you'll find that the calculation using 200 data points will more than likely have smaller error, in fact, the error will be proportional to the step size, here being 1/25 for the 50 step mesh, and 1/100 for the 200 step mesh, so the error should be reduced by a factor of four.

The same process applies to 'improved euler' as far as calculating the error goes, however, the factor of error may be different, though I wouldn't be able to say for sure since I don't know what the 'improved euler' method is.

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