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Chem Differential Eq HW Solutions Fall 2011 168

Chem Differential Eq HW Solutions Fall 2011 168 - A168...

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A168 Appendix A Ordinary Differential Equations: Review of Concepts and Methods 13. An integrating factor is e x 2 2 , so e x 2 2 y + xe x 2 2 y = xe x 2 2 d dx e x 2 2 y = e x 2 2 x ye x 2 2 = xe x 2 2 dx = e x 2 2 + C y = 1 + Ce - x 2 2 . We now use the initial condition: y (0) = 0 0 = 1 + C C = - 1 y = 1 - e - x 2 2 . 17. An integrating factor is sec x (see Exercise 9), so sec xy + y tan x sec x = tan x sec x d dx [ y sec x ] = sec x tan x y sec x = tan x sec xdx = sec x + C y = 1 + C cos x. We now use the initial condition: y (0) = 1 1 = 1 + C C = 0 y = 1 . 21. (a) Clear. (b) e x as a linear combination of the functions cosh x , sinh x : e x = cosh x +sinh x . (c) Let a,b,cd be any real numbers such that ad - bc = 0. Let y 1 = ae x + be - x and y 2 = ce x + de - x . Then y 1 and y 2 are solutions, since they are linear combinations of two solutions. We now check that
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