4 15 pts suppose is a solution of the homogeneous

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______________________________________________________________________ 4. (15 pts.) Suppose is a solution of the homogeneous second order linear equation Very neatly obtain the recurrence formula(s) needed to determine the coefficients of y ( x ). DO NOT WASTE TIME ATTEMPTING TO GET THE NUMERICAL VALUES OF THE COEFFICIENTS. First, for all x near zero. From this you can deduce that we have
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TEST3/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) (a) Suppose that f ( t ) is defined for t 0. What is the definition of the Laplace transform of f in terms of a definite integral?? for all s for which the integral converges. (b) Using only the definition, not the table, compute the Laplace transform of ______________________________________________________________________ 6. (10 pts.) Compute when (a) (b) ’Tis just the usual magic of multiplication by ’1’ in the correct form or the addition of ’0’ suitably transmogrified with linearity tossed into the mix. ______________________________________________________________________ 7. (5 pts.) Locate and classify the singular points of the following second order homogeneous O.D.E. Use complete sentences to describe the type of points and where they occur. An equivalent equation in standard form is From this, we can see easily that x 0 = 0 is an irregular singular point of the equation, and x 0 = 4 is a regular singular point. All other real numbers are ordinary points of the equation.
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TEST3/MAP2302 Page 4 of 4 ______________________________________________________________________ 8. (15 pts.) Using only the Laplace transform machine, solve the following first order initial-value problem. By taking the Laplace Transform of both sides of the differential equation, and using the initial condition, we have By taking inverse transforms now, we quickly obtain ______________________________________________________________________ 9. (10 pts.)
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