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Cauchy differential equation

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Prove d/dt(u(x(t),t))+tanh(x(t))(d/dx(u(x(t),t)))=0
u(x(t),0)=a(x(t))
limit as x tends to infinity of u(x,t)=0

has at most one solution.Explain why there is no boundary condition at x=0
and find a solution for the special case a(x(t))=sinh(x(t))

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Solution Summary

This shows why a given differential equation has at most one solution.

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Polynomials and Differentials: The Cauchy Riemann Equation

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Please help solve this attached problem that involves polynomials and differentials. It involves an application of the Cauchy Riemann equation.

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