Solve the equation by non-trivial method

Question

Non-Trivial Solution

Solve the following problem:

∇²T(r,0) = (∂²/∂r + 1/r ∂/∂r+1/r² ∂²/∂Θ²)T(r,Θ)=0 For 0
Subject to T_Θ (r,0) = 0,T(r,π/ 2)=0,T(1,Θ)=1

Show that the equation y′+λ x²y=0

Can posses a nontrivial solution which vanishes at x=0 and x=1 only if λ is such that J_1/4 (1/2√λ)=0,and that corresponding to such a characteristic number λ_k, Any multiple of the function Φ_k(x)= √xJ_1/4(1/2√λ_kx² is a solution with the required properties.