Query 15 Differential Equations

Question:  Suppose y1 and y2 are solutions to y '' + 2 t y ' + t^2 y = 0.  If y1(3) = 0, y1 ' (3) = 0, y2(3) = 1 and y2 ' (3) = 2, can you say whether {y1, y2} is a fundamental set?  If so, is it or isn't it?

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Question:  Are y1 = 2 e^(-2 t) cos(t) and y2 = e^(-2 t) sin(t) solutions to the equation

y '' + 4 y ' + 5 y = 0?

What are the initial conditions at t = 0?

Is {y1, y2} a fundamental set?

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Question:  y1_bar = 2 y1 - 2 y2 and y2_bar = y1 - y2.  Is {y1_bar, y2_bar} a fundamental set?

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Given Solution:  Note that y_1_bar = 2 * y_2_bar. 

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Question:  Is {e^t, 2 e^(-t), sinh (t) } a fundamental set on the interval (-infinity, infinity)?

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