equ1 = DSolve[X''[x] + \[Lambda]*X[x] == 0, X[x], x]; X[x] = X[x] /. equ1[[1]]; equ2 = Simplify[X[x] /. {x -> 0}, \[Lambda] > 0] == 0; equ3 = Simplify[X[x] /. {x -> L}, \[Lambda] > 0] == 0; equ4 = Simplify[equ3, equ2 == 0]; equ5 = Reduce[Sin[L*\[Lambda]^(1/2)] == 0, \[Lambda]]; equ6 = Simplify[ equ5, {L > 0 && \[Lambda] > 0 && C[1] \[Element] Integers}]; equ7 = Simplify[Solve[equ6, \[Lambda]]]

Solve[equ6[[1]][[2]], \[Lambda]]~Join~Solve[equ6[[2]][[2]], \[Lambda]]