I need to solve a linear, second order, homogeneous ODE, and I'm using the Frobenius method. This amounts to setting:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]y = \sum_{n=0}^{\infty} c_n x^{n+k} [/tex]

then getting y' and y'', plugging in, combining like terms, and setting the coefficient of each term to 0 to solve for the c_{n}'s. This will give solutions in the form of infinite power series. Specifically, setting the first coefficient to 0 will result in a quadratic equation in k, which is then solved to get the starting powers for the two linearly independent solutions. However, in a problem I'm doing, which I can post if anyone wants, I get imaginary values for k. Is this valid? It means the series will have the form:

[tex]y = c_0 x^i + c_1 x^{i+1} + c_2 x^{i+2} + ...[/tex]

With the c_{n}complex. Will this converge to a real, finite number for all x when I plug in real boundary conditions?

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# Frobenius method with imaginary powers

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