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    Fundamental Set of Solutions to an ODE

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    Suppose that p and q are continuous on some open interval I, and suppose that y1 and y2 are solutions of the ODE
    y'' + p(t)y' + q(t)y = 0
    on I.
    (a) Suppose that y1, y2 is a fundamental set of solutions. Prove that z1, z2, given by
    z1 = y1 + y2, z2 = y1 − y2,
    is also a fundamental set of solutions.

    (b) Prove that if y1 and y2 achieve a maximum or a minimum at the same point in I,
    then they cannot form a fundamental set of solutions on this interval.

    (c) Prove that if y1 and y2 form a fundamental set of solutions on I, then they cannot
    have a common inflection point in I, unless p and q are both 0 at this point.
    (d) If 0 is in I, show that y(t) = t^3 cannot be a solution of the ODE on I.

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    https://brainmass.com/math/calculus-and-analysis/fundamental-set-solutions-ode-170921

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    (a) Suppose that y1, y2 is a fundamental set of solutions. Prove that z1, z2, given by , is also a fundamental set of solutions.

    When the two solutions are linearly dependent we have for some constant c:

    Then:

    So a condition for independence is that the Wronskian W(x) is non zero:

    In our case we can write:

    Therefore:

    Thus form an independent fundamental solutions set as well.

    A little theoretical detour:

    The equation is:

    The two ...

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