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The solution ex, e-x, and e2x of 
will be:
will be:- a)linearly dependent for x ϵ [-2, 2] and linearly independent elsewhere.
- b)linearly independent for x ϵ [-1, 1] and linearly independent elsewhere.
- c)linearly independent on every real interval.
- d)linearly dependent for all real x.
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
The solution ex, e-x, and e2xofwill be:a)linearly dependent for x [-2...
Write auxillary equation by replacing
Equate to 0 and solve for (m) (f(m) = 0)
for m1, m2, m3, … (real and different roots)
Complementary function (CF) is given as:
Calculation:
The above expression is in the form f(D)y = X and can be written as:
D3 y - 2D2y - Dy + 2y = 0
y(D3 - 2 × D2 - D + 2) = 0
D3 - 2D2 - D + 2 = 0
The auxiliary equation is f(m) = 0
m3 - 2m2 - m + 2 = 0
(m - 1)(m + 1)(m - 2) = 0
So, the roots the auxiliary equation are m1 = 1, m2 = -1, m3 = 2.
The characteristics of the roots are real and distinct.
C.F = yc = c1em1x + c2em2x + c3em3x
yc = c1ex + c2e-x + c3e2x
So the solution will be linearly independent on every real interval.
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The solution ex, e-x, and e2xofwill be:a)linearly dependent for x [-2, 2] and linearly independent elsewhere.b)linearly independent for x [-1, 1] and linearly independent elsewhere.c)linearly independent on every real interval.d)linearly dependent for all real x.Correct answer is option 'C'. Can you explain this answer?
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