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Fourier transform Maxwell's equations, solve the system of ODEs for the electric and magnetic fields, and apply inverse Fourier transforms to find the fields in real space.

Fourier transform the Helmholtz equation, solve the resulting equation in the Fourier domain, then find the solution via inverse Fourier transform.

Fourier transform the diffusion equation, incorporate the source term in the transformed domain, solve for the transformed solution, and inverse Fourier transform.

Fourier transform Laplace's equation, solve the algebraic equations, then apply the inverse Fourier transform to obtain the solution in the spatial domain.

Apply Fourier transform to both sides, solve the resulting ODE, apply inverse Fourier transform to find the solution.

Apply Fourier transform to the KdV equation, solve the resulting nonlinear ODE, use inverse scattering transform if needed, then inverse Fourier transform to real space.

Apply the Fourier transform to both sides of Poisson's equation, solve the resulting algebraic equation in Fourier space, then inverse Fourier transform to get the potential field.

Fourier transform the equation, find general solutions to the resulting ODEs, use initial conditions to find specific solutions, inverse Fourier transform back.

Fourier transform Schrodinger's equation, solve the resulting ODE with given potential, and apply boundary conditions after inverse Fourier transforming.