Example 5 Given: $a, b, c$ are all positive numbers, find the maximum value of $y=\frac{a b+2 b c}{a^{2}+b^{2}+c^{2}}$.
Analysis In the process of using the arithmetic mean inequality, note that the coefficient ratio of $a b$ and $b c$ should be $1: 2$, so we use parameters to determine the method of splitting $b^{2}$.
Let $\lambda \in(0,1)$, then $a^{2}+\lambda b^{2} \geqslant 2 \sqrt{\lambda} a b, (1-\lambda) b^{2}+c^{2} \geqslant 2 \sqrt{1-\lambda} b c$.
Let $2 \sqrt{\lambda}: 2 \sqrt{1-\lambda}=1: 2$, we get $\lambda=\frac{1}{5}$,
So $a^{2}+\frac{1}{5} b^{2} \geqslant 2 \sqrt{\frac{1}{5}} a b, \frac{4}{5} b^{2}+c^{2} \geqslant 4 \sqrt{\frac{1}{5}} b c$.
So $a^{2}+b^{2}+c^{2} \geqslant 2 \sqrt{\frac{1}{5}}(a b+2 b c)$.
So $\frac{a b+2 b c}{a^{2}+b^{2}+c^{2}} \leqslant \frac{a b+2 b c}{2 \sqrt{\frac{1}{5}}(a b+2 b c)}=\frac{\sqrt{5}}{2}$.
So the maximum value is $\frac{\sqrt{5}}{2}$.