de-t2-a

# To obtain a 4th order homogeneous linear constant

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(a) It is trivial to obtain a 4th order homogeneous linear constant coefficient ordinary differential equation with f and g as solutions. Do so. (b) It’s only slightly messier to obtain a 2nd order homogeneous linear ordinary differential equation with { f , g } as a fundamental set of solutions. Do so. Hints: (a) Here one should expect that a fundamental set of solutions should be { 1, x , sin( x ), cos( x ) }, arising from the auxiliary equation m 2 ( m 2 + 1) = 0. The constant coefficient equation has a family resemblance. (b) Substitute f and g into the ODE y p ( x ) y q ( x ) y 0 to obtain a linear system in p ( x ) and q ( x ). Find p and

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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: m ( m-2 π ) 2 (m - (5i)) 3 (m - (-5i)) 3 =0 (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? y c 1 c 2 e 2 π x c 3 xe 2 π x c 4 sin(5 x ) c 5 cos(5 x ) c 6 x sin(5 x ) c 7 x cos(5 x ) c 8 x 2 sin(5 x ) c 9 x 2 cos(5 x ) The order of the differential equation is 9. ______________________________________________________________________ 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.] (*) . x 2 2 y 1 ce 2 y (a) Differentiating (*) with respect to x and then replacing c yields 2 x 2 dy dx 2 ce 2 y dy dx dy dx 2 2( x 2 2 y 1) e 2 y e 2 y dy dx 4 y 2 x 2 . Thus, a differential equation for the family of curves is given by dy dx x 2 y x 2
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to obtain a 4th order homogeneous linear constant...

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