Question

Find the periodic solutions of the differential equations \((a) \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{ky}=\mathrm{f}(\mathrm{x}),(b) \frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{d}^{3} \mathrm{x}}+\mathrm{ky}=\mathrm{f}(\mathrm{x})\)

where \(k\) is a constant and \(f(x)\) is a \(2 \pi\) - periodic function.

Consider a Fourier series expansion for \(f(x)\) using the complex form, \(f(x)=\sum_{n=-\infty}^{n=+\infty} f_{n} e^{i n x}\) and try a solution of the form \(y(x)=\sum_{n=-\infty}^{n=+\infty} y_{n} e^{i n x}\)

Answer 1

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