User:EGM6321.f12.team7.Marjanovic/report6/6.2

=R*6.2=

Problem 2: Solving a L2-ODE Using Method 2
Based on lecture notes Section 31

Given: A L2-ODE With Boundary Conditions
Boundary Conditions:

Trial Solution:

Find: Solve the L2-ODE Using Trial Solution
Solve the given homogeneous L2-ODE using the trial 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, which can be easily shown to be:

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

Since $$\displaystyle x^r $$ can not equal zero for all values of r,

Factoring the expression, we get:

Thus, the general solution will look like:

Using the boundary conditions from ($$) and ($$),

Solving the system of equations using substitution, we get:

Finally, we substitute the constants back in to ($$)