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Solving a second order differential equation using Fourier s

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Consider the ODE

y" - a^2 * y = H(x-Pi/2)


Where H is the step function.

solve for y using Fourier series
The Fourier series for the step function exhibits the Gibss phenomenon. Will the solution y(x) exhibit it too? explain why or why not.

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This is an exhaustive 13 pages solution that contains all the details and analysis needed to solve the differential equation.