Evaluate the limit along the stated paths, or type "DNE" if the limit Does Not Exist:
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}
\]
Along the path $x=0$ :
Along the path $y=0$ :
Along the path $y=4 x$ :

Step 1 ：First, we substitute $x=0$ into the equation, we get $\lim _{y \rightarrow 0} \frac{y^{3}}{y^{2}}=\lim _{y \rightarrow 0} y=0$

Step 2 ：Then, we substitute $y=0$ into the equation, we get $\lim _{x \rightarrow 0} \frac{0}{x^{2}}=0$

Step 3 ：Next, we substitute $y=4x$ into the equation, we get $\lim _{x \rightarrow 0} \frac{4x^{2}+64x^{3}}{16x^{2}+x^{2}}=\lim _{x \rightarrow 0} \frac{4x+64x^{2}}{17x}=\lim _{x \rightarrow 0} \frac{4+64x}{17}=\frac{4}{17}$

Step 4 ：Since the limits along the paths $x=0$, $y=0$ and $y=4x$ are not the same, the limit does not exist

Step 5 ：So, the final answer is \(\boxed{\text{DNE}}\)