# Above is a typographical error a 5 pts write down the

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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? The order of the differential equation is 8. ______________________________________________________________________ 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.] (*) . (a) Differentiating (*) with respect to x and then replacing c using (*) yields a differential equation for the family of curves. (b) An ODE for the orthogonal trajectories is now given by or equivalently, . (c) This little equation is equivalent to one that is linear with x as the dependent variable and y the independent variable(!): This linear equation has μ = exp(y) as an integrating factor. Using the standard recipe, a one-parameter family of solutions is given by .

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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
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