If xe -x is an element of a solution set for a constant coefficient linear homog

Question

If xe-x is an element of a solution set for a constant coefficient linear homogeneous ODE, then a factor of the associated characteristic equation would be Question 16 options: A) a4 + 2a2 + 1 B) a2 + 2a + 4 C) a2 - 2a + 2 D) a2- 1 E) a2 - 2a + 1 F) a2 + 2a + 1 G) (a2 + 2a + 2)2 H) none of these If xe-x is an element of a solution set for a constant coefficient linear homogeneous ODE, then a factor of the associated characteristic equation would be If xe-x is an element of a solution set for a constant coefficient linear homogeneous ODE, then a factor of the associated characteristic equation would be A) a4 + 2a2 + 1 B) a2 + 2a + 4 C) a2 - 2a + 2 D) a2- 1 E) a2 - 2a + 1 F) a2 + 2a + 1 G) (a2 + 2a + 2)2 H) none of these A) a4 + 2a2 + 1 B) a2 + 2a + 4 C) a2 - 2a + 2 D) a2- 1 E) a2 - 2a + 1 F) a2 + 2a + 1 G) (a2 + 2a + 2)2 H) none of these A) a4 + 2a2 + 1 B) a2 + 2a + 4 C) a2 - 2a + 2 D) a2- 1 E) a2 - 2a + 1 F) a2 + 2a + 1 G) (a2 + 2a + 2)2 H) none of these

Explanation / Answer

If y = (c1)xe^(-x) is a solution of the diff eq, the fact that you have (c1)xe^(ax) instead of just (c1)e^(ax) indicates you have a repeated root. The general solution form is:

y = (c1)xe^(-x) + (c2)e^(-x).

The repeated root is a = -1 so the characteristic equation is:

(a + 1)(a + 1) = 0

a^2 + 2a + 1 = 0

which is choice F.

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