Separation constant

"The partial differential equation a sin 2 ⁡ ( θ ) Φ ( ϕ ) + Φ ( ϕ ) cos ⁡ ( θ ) ∂ Θ ( θ ) ∂ θ + ∂ 2 Φ ∂ ϕ 2 = 0 {\displaystyle a\sin ^{2}(\theta )\Phi (\phi )+\Phi (\phi )\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}+{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}=0} can be rewritten a sin 2 ⁡ ( θ ) + cos ⁡ ( θ ) ∂ Θ ( θ ) ∂ θ = − 1 Φ ( ϕ ) ∂ 2 Φ ∂ ϕ 2 ≡ K {\displaystyle a\sin ^{2}(\theta )+\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}=-{\frac {1}{\Phi (\phi )}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\equiv K} , where K {\displaystyle K} is a separation constant that depends on neither θ {\displaystyle \theta } nor ϕ {\displaystyle \phi } . This then yields two ordinary differential equations."