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Thread: System of differential equations

  1. #1
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    System of differential equations

    Let x'=Ax be a system of differential equations, where A is a real constant 3x3 matrix with exactly two different real eigenvalues.

    which one of the following statements must holdonly one of them follows from the data)

    a)there is no base of solutions in the form V(e^at) where a is a real number and V is a real vector

    b)there is a solution in the form V(e^at) where a is a real number and V is a real vector

    I think that the two follows

    Thank's in advance
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  2. #2
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    Re: System of differential equations

    I would suggest a). Because two distinct real eigenvalues implies that one of them is repeated (because it's a 3x3 matrix, not a 2x2) and so I expect a solution of the form
    $$\vec X = c_1\vec{V_1}e^{a_1t} + c_2\vec{V_2}e^{a_2t} + c_3t\vec{V_3}e^{a_2t}$$
    Mind you, if $c_2 = c_3 = 0$ then you have the given form I suppose.
    Last edited by Archie; Aug 12th 2019 at 05:41 AM.
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  3. #3
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    Re: System of differential equations

    Yes but the first statement is also true
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  4. #4
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    Re: System of differential equations

    Yes, there is no basis of all solutions with only one exponential term. But is there any reason why $$\vec X = c_1\vec{V_1}e^{a_1t}$$ wouldn't be a solution for every value of $c_1$? In this case, the given expression is a basis for a subset of all possible solutions.
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