This is the product solution of the two independent solutions of the form:
R(r)Θ(θ)
where R(r) is the radial component and Θ(θ) is the angular component.
We can assume that the second term of the equation is separable, meaning that we can rewrite it as:
∂2u
∂θ2
= -λu
where λ is a separation constant.
Substituting this into the original equation, we get:
∂2u
∂r2
1
r
∂u
∂r
λ
r
2
u
= 0
Multiplying both sides by r^2, we get:
r^2
∂2u
∂r2
r
∂u
∂r
λu
= 0
This is a form of the Bessel differential equation, which has solutions of the form:
u(r) = AJv(√λr) + BYv(√λr)
where Jv and Yv are Bessel functions of the first and second kind, respectively, and A and B are constants determined by the boundary conditions.
Substituting this back into the product solution, we get:
Dr(r, θ) = (d^2/dt^2 + λ) [R(r)Θ(θ)]
This shows that the product solution is a linear combination of functions that satisfy the partial differential equation.