3. Let $\lambda>0$, find the largest constant $c=c(\lambda)$, such that for all non-negative real numbers $x, y$, we have $x^{2}+y^{2}+\lambda x y \geqslant c(x+y)^{2}$.
3. For $\lambda$, consider two cases:
(i) When $\lambda \geqslant 2$, we have $x^{2}+y^{2}+\lambda x y \geqslant x^{2}+y^{2}+2 x y=(x+y)^{2}$. The equality holds when $x y=0$.
(ii) When $0<\lambda<2$, we have $x^{2}+y^{2}+\lambda x y=(x+y)^{2}-(2-\lambda) x y \geqslant$ $(x+y)^{2}-(2-\lambda)\left(\frac{x+y}{2}\right)^{2}=\frac{2+\lambda}{4}(x+y)^{2}$. The equality holds when $x=y$.
Therefore, the largest constant $c(\lambda)=\left\{\begin{array}{l}1, \lambda \geqslant 2, \\ \frac{2+\lambda}{4}, 0<\lambda<2 .\end{array}\right.$