Particular integral y p of the ode warning a if you

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particular integral, y p , of the O.D.E. [ Warning: (a) If you skip a critical initial step, you will get no credit!! (b) Do not waste time attempting to find the numerical values of the coefficients!! ]
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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? ______________________________________________________________________ 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.] (*) .
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TEST2/MAP2302 Page 4 of 4 ______________________________________________________________________ 7. (10 pts.) The nonzero function f ( x ) = exp( x) is a solution to the homogeneous linear O.D.E. (*) (a) Reduction of order with this solution involves making the substitution into equation (*) and then letting w = v . Do this substitution and obtain in standard form the first order linear homogeneous equation that w must satisfy. (b) Finally, obtain an integrating factor, μ , for the first order linear ODE that w satisfies and then stop. Do not attempt to actually find v. _________________________________________________________________ Silly 10 Point Bonus: Suppose that the function , where n is a positive integer, is a solution to the constant coefficient homogeneous linear equation (*) where m > n . What can you say about the coefficients of the ODE, and what can you say about its fundamental set of solutions?? Why??? [Say where your work is, for it won’t fit here.]
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