For λ ∈ ℝ, consider the boundary value problem
\(\left.\begin{array}{l} x^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}+λ y=0, x ∈[1,2] \\ y(1)=y(2)=0 \end{array}\right\}-\left(P_{λ}\right)\)
Which of the following statement is true?
1
There exists a λ0 ∈ ℝ such that (Pλ) has a nontrivial solution for any λ > λ0
2
{λ ∈ ℝ : (Pλ) has a nontrivial solution} is a dense subset of ℝ.
3
For any continuous function f: [1,2] → ℝ with f(x) ≠ 0 for some x ∈ [1,2], there exists a solution u of (Pλ) for some λ ∈ ℝ such that \(\int_1^2 fu \ne 0\)
4
There exists a λ ∈ ℝ such that (Pλ) has two linearly independent solutions.
5
Question Not Attempted