Let the Laplace equation ∇2u(r, θ) = 0 be satisfied inside the disk

1.	Let the Laplace equation ∇2u(r, θ) = 0 be satisfied inside the disk r < 5, where r and θ are the polar coordinates,and let the boundary condition be u(5, θ) = sin2θ. Find u(r, θ).
Answer» sin²θ = (e^iθ − e^−iθ)²/- 4 = 1 − cos2θ/2. Thus u(r, θ) = (1 − r²cos2θ/25)/2.

Let the Laplace equation ∇2u(r, θ) = 0 be satisfied inside the disk r < 5, where r and θ are the polar coordinates,and let the boundary condition be u(5, θ) = sin2θ. Find u(r, θ).

Answer»

sin²θ = (e^iθ − e^−iθ)²/- 4 = 1 − cos2θ/2.

Thus

u(r, θ) = (1 − r²cos2θ/25)/2.

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Let the Laplace equation ∇2u(r, θ) = 0 be satisfied inside the disk r &lt; 5, where r and θ are the polar coordinates,and let the boundary condition be u(5, θ) = sin2θ. Find u(r, θ).

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Let the Laplace equation ∇2u(r, θ) = 0 be satisfied inside the disk r < 5, where r and θ are the polar coordinates,and let the boundary condition be u(5, θ) = sin2θ. Find u(r, θ).