# 3 12 pts without evaluating any integrals and using

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Write the solution to the IVP in piecewise-defined form. ______________________________________________________________________ 3. (12 pts.) Without evaluating any integrals and using only the table provided, properties of the Laplace transform, and appropriate function identities, obtain the Laplace transform of each of the functions that follows. (4 pts./part) (a) (b) (c)

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MAP2302/FinalExam Page 4 of 6 ______________________________________________________________________ 4. (10 pts.) Obtain the recurrence formula(s) satisfied by the coefficients of the power series solution y at x 0 = 0, an ordinary point of the homogeneous ODE, ______________________________________________________________________ 5. (10 pts.) (a) (4 pts.) Obtain the differential equation satisfied by the family of curves defined by the equation (*) below. (b) (3 pts.) Next, write down the differential equation that the orthogonal trajectories to the family of curves defined by (*) satisfy. (c) (3 pts.) Finally, solve the differential equation of part (b) to obtain the equation(s) defining the orthogonal trajectories. (*) y e cx
MAP2302/FinalExam Page 5 of 6 ______________________________________________________________________ 6. (10 pts.) The equation has a regular singular point at x 0 = 0. Substitution of into the ODE and a half a page of algebra yields Using this information, (a) write the form of the two linearly independent solutions to the ODE given by Theorem 6.3 without obtaining the numerical

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