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Mth 279

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Query 27 question

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In Query 27, this is the second to last question

Question:  Find all values of mu such that any fundamental set [ y_1, y_2 ] of the system

y ' = [1, 3; mu, -2] y

have the property that the limit of the expression (y_1(t))^2 + (y_2(t))^2, as t -< infinity, is zero.

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I'm very unsure of where to start. Normally you would have to find the wronskian and show that it is non zero. But I can't find a value of W because I don't know mu. Also, when you state ... the limit of the expression (y_1(t))^2 + (y_2(t))^2... Are y1(t) and y2(t) vectors? Scalars?

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We assume solution vector

`y = [ y_1, y_2 ].

So

`y

is a vector with two components y_1 and y_2, both components being scalar functions of t.

These functions will approach zero as t approaches infinity provided they are both of the form

e^(-k t) * h(t),

with positive k (so that -k is negative), where h(t) could be a sine or cosine function, or a constant function.

So you are looking for the values of mu that give you exponentials with negative exponents, or whose real parts are negative.

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question form

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In query 27, this is the last question

  A particle moves in an unspecified force field in such a way that its position vector r(t) = x(t) i + y(t) j and the corresponding velocity vector v(t) = r ' (t) satisfy the equationv ' = 2 k X v

Write this condition as a system

v ' = A v,

with v = [v_x; v_y].

If the particle starts at position r(0) = 2 i + j, v(0) = i + 2 j, find its position at t = 3 pi / 2.

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what does the X stand for in the equation v ' = 2 k X v? And would I set up a vector that looks like v = [r'; v']?

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`k X `v is the cross product of the vectors `k and `v.

See also the example in your text.

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