The ode of problem c is linear as written but it is

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The ODE of problem (c) is linear as written, but it is not in standard form with the coefficient on the derivative being one. Here is the standard form version: It has an obvious integrating factor of μ = x -2 for x > 0. Multiplying the ODE by μ , integrating, and then dealing with the initial condition allows you to produce x -2 y = x 2 - 2, an implicit solution. An obvious explicit solution is y ( x ) = x 4 - 2 x 2 , actually defined for all x . (d) First, the corresponding homogeneous equation is , which has an auxiliary equation given by 0 = m 2 + 1. Thus, m = ± i , and a fundamental set of solutions for the corresponding homogeneous equation is . Obviously the driving function here is a UC function. Thus, we may use the UC machinery to nab a particular integral for the ODE. A UC set is given by If , and this is a solution to the ODE of (d), then Solving the system yields A = 4, B = 0, and C = -8. Thus, a particular integral of the ODE above is Consequently, the general solution is
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MAP2302/FinalExam Page 3 of 6 ______________________________________________________________________ 2. (8 pts.) Suppose that the Laplace transform of the solution to a certain initial value problem involving a linear differential equation with constant coefficients is given by 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, First, From this you can deduce that c 2 = c 1 /2 , and that for n 1, we have ______________________________________________________________________ 5. (10 pts.) (a) (4 pts.)
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