User:EGM6321.f12.team7.Marjanovic/report6/6.8

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Problem 2: Roots of a Trial Solution
Based on lecture notes Section 35

Given: Homogeneous L2-ODE
For the above equation, using a trial solution of:

and a characteristic equation of:

We are given the two roots of the equation as:

Find: Validity of the Root and Solution
Explain why ($$) is not a valid root, i.e., $$\displaystyle u_2(x) = e^{xr_2(x)} $$ is not a valid solution.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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To start off, we must find the derivatives of the trial solution

Now substituting these derivatives and the trial solution in to ($$)

From here it is quite obvious that:

And thus, ($$) is not a valid root.