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Calculus Level 5

The function F(x,y,z)F(x, y, z) is defined in the following manner:

F(x,y,z)=2F(x,y,z), where 2 is the Laplacian.F(x, y, z) = \nabla^2 F(x, y, z) \text{, where } \nabla^2 \text{ is the Laplacian.}

The solution to this can be written in the following form:

F(x,y,z)=a(u)+a(u), where u=x+y+z.F(x, y, z) = a(u) + a'(u) \text{, where } u = x+y+z.

The function a(u)a(u) obeys the nonlinear differential equation a(2u)=2a(u)a(u)a(2u) = 2 a(u) \cdot a'(u). The average of the possible values of a(0)a(0) can be written as ab-\frac{a}{b}, where aa and bb are coprime positive integers. What is the value of a+ba+b?

It may help to read about the Laplacian here.

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