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Differential Equation

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Evaluate the fundamental solution of the equation u^(4) - 2u'' + u. More in general, evaluate the fundamental solution of u^(4) - (m+1)u'' + mu, where m>=0.

Please note that you are not solving the equation set equal to zero. Since you are looking for the fundamental solution you set the equation equal to the delta distribution delta_0. So you have to use the Fourier Transform in distribution sense.

https://brainmass.com/math/calculus-and-analysis/differential-equation-fourier-transform-distribution-349951

Solution Preview

To find fundamental solution we need to solve the following equation for function :

We take Fourier transform of both sides of the equation.
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Recall the property of Fourier Transform: . Hence denoting , and using the fact that Fourier transform of delta-function is we ...

Solution Summary

This posting uses the Fourier Transform in distribution sense.

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