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For $\lambda \in \mathbb{R}$, consider the boundary value problem $$\left. \begin{matrix}  x^2 \frac{d^2y}{dx^2}+2x\frac{dy}{dx}+\lambda y=0, & x\in [1,2] \\ y(1)=y(2)=0 & \end{matrix} \right \rbrace \quad - (P_\lambda)$$ Which of the following statement is true?

  1. There exists a $\lambda_0 \in \mathbb{R}$ such that $(P_\lambda )$ has a nontrivial solution for any $\lambda >\lambda_0$
  2.  $ \lambda \in \mathbb{R} : (P_\lambda ) $  has a nontrivial solution is a dense subset of $\mathbb{R}$
  3. For any continuous function $f:[1,2] \rightarrow \mathbb{R}$ with $f(x) \neq 0$ for some $x\in [1,2]$, there exists a solution $u$ of $(P_\lambda )$ for some $\lambda \in \mathbb{R}$ such that $\int_{1}^{2}fu\neq 0$
  4. There exists a $\lambda \in \mathbb{R}$ such that $(P_\lambda)$ has two linearly independent solutions. 
in Differential Equation by Expert (2.4k points)

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