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    Linear dependence of solutions

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    One solution to ty"-(t+2)y'+2y=0 is exp(t)
    Find a second linearly independent solution.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:06 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/linear-dependence-solutions-28803

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    SOLUTION This solution is FREE courtesy of BrainMass!

    Hi there!

    Here is the solution in two file formats as usual.
    I'm sorry it took me a while, but I wanted to show you two different methods how to approach this kind of problem (who know's in a test it might come handy).

    First Method: Wronskian

    Write the equation as:

    The two solutions must satisfy:

    Multiplying equation (2) by y1 and subtracting equation (1) multiplied by y2 we obtain:

    Using the Wronskian's definition:

    We see that:

    So:

    This is a first order separable homogenous differential equation and we solve it for W:

    Where W0 is just a constant of integration.

    So rewriting the equation

    As:

    We see that

    And:

    Thus the Wronskian is:

    Now, from the definition of the Wronskian we also get:

    Which in our case, translates to:

    We can solve this integral using integration by parts twice:

    Thus (remember that W0 is just an arbitrary constant to be determined by initial conditions so we can absorb the negative sign into the constant):

    We can confirm this solution by substituting it back into the equation:

    So the solution:

    Is the second linearly independent solution to this ODE.

    Second method: Reduction of order.

    Since we know one solution, we assume a second solution of the form:

    Thus:

    Substituting this back into the equation:

    Since is a solution the term in the last curly parenthesis is zero.

    So we are left with:

    But here we have only terms of and , so this is actually a first order equation for .

    Using the equation becomes:

    Substituting we simplify the equation further:

    This is a simple separable equation:

    But:

    Therefore (We calculated this integral before):

    And since we have defined we get:

    Maple Verification:

    > eqn:=t*diff(diff(y(t),t),t)-(t+2)*diff(y(t),t)+2*y(t)=0;

    > dsolve(eqn);

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:06 pm ad1c9bdddf>
    https://brainmass.com/math/linear-algebra/linear-dependence-solutions-28803

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