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However this gives no solution (over-specified):
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Now a counter-productive transform:
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You would now have an 8th order ODE to solve.
Attempting to use the fourier transform with initial conditions yields no solution:
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The following ODE has different solutions depending on whether 'pointwise'=true has been specified.
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So the solution as provided does not satisfy the initial conditions in a pointwise fashion, as can be seen from
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To instead enforce the conditions to be pointwise satisfied, the following can be used:
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