University of Florida/Egm4313/s12.team11.perez.gp/R3.2

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Problem Statement[edit]

Developing the second homogeneous solution for the case of double real root as a limiting case of distinct roots.

Given[edit]

Consider two distinct roots of the form:

and

(where is perturbation).

Part 1[edit]

Given[edit]

Find the homogeneous L2-ODE-CC having the above distinct roots.

Solution[edit]


(1)

Part 2[edit]

Given[edit]

Show that is a homogeneous solution. (2)

Solution[edit]

Let's find the corresponding derivatives:

If we now take these three equations and plug them into the homogeneous L2-ODE-CC (1), we get:

Since the left and right hand sides of the equation are zero, the solution is in fact a homogeneous equation.

Part 3[edit]

Given[edit]

Find the limit of the homogeneous solution in (2) as epsilon approaches zero (think l'Hopital's Rule).

Solution[edit]

Using l'Hopital's Rule,

(this is an indeterminate form).

L'Hopital's Rule states that we can divide this function into two functions, and , and then find their derivatives and attempt to find the limit of . If a limit exists for this, then a limit exists for our original function.

Part 4[edit]

Given[edit]

Take the derivative of with respect to lambda.

Solution[edit]

Taking the derivative with respect to lambda, we find that:

.

It is important to remember that we must hold as a constant when finding this derivative.

Part 5[edit]

Given[edit]

Compare the results in parts (3) and (4), and relate to the result by using variation of parameters

Solution[edit]

Taking a closer look at Parts 3 and 4 of this problem, we discover that they're in fact equal:

Part 6[edit]

Given[edit]

Numerical experiment: Compute (2) setting lambda equal to 5 and epsilon equal to 0.001 </math>, and compare to the value obtained from the exact second homogeneous solution.

Solution[edit]

After performing these calculations, from (2) we get 148.478.

And from the exact second homogeneous solution, we get 200.05.