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Unformatted text preview: So in this case, we would not have a function identity on any interval of the line. It follows that a k = 0 for every k with 0 ≤ k ≤ n . Consequently, the fundamental set of solutions includes the following: TEST2/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) The factored auxiliary equation of a certain homogeneous linear O.D.E. with real constant coefficients is as follows: Note: The plus sign in the factor ( m + ( π i )) 2 above is a typographical error. (a) (5 pts.) Write down the general solution to the differential equation. [ WARNING: Be very careful. This will be graded Right or Wrong!! ] (b) (5 pt.) What is the order of the differential equation? The order of the differential equation is 8. ______________________________________________________________________ 6. (15 pts.) (a) Obtain the differential equation satisfied by the family of curves defined by the equation (*) below. (b) Next, write down the differential equation that the orthogonal trajectories to the family of curves defined by (*) satisfy. (c) Finally, solve the differential equation of part (b) to obtain the equation(s) defining the orthogonal trajectories. [These, after all, are another family of curves.] (*) . (a) Differentiating (*) with respect to x and then replacing c using (*) yields a differential equation for the family of curves. (b) An ODE for the orthogonal trajectories is now given by or equivalently, ....
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 Fall '08
 STAFF
 Vector Space

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