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de-t2(1)

# B next write down the differential equation that the

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(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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b Next write down the differential equation that the...

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