For a twice continuously differentiable function g: ℝ → ℝ, define
\(u_g(x, y)=\frac{1}{y} \int_{-y}^y g(x+t) d t \quad \text { for }(x, y) \in \mathbb{R}^2, \quad y>0\)
Which one of the following holds for all such g?
1
\(\frac{\partial^2 u_g}{\partial x^2}=\frac{2}{y} \frac{\partial u_g}{\partial y}+\frac{\partial^2 u_g}{\partial y^2}\)
2
\(\frac{\partial^2 u_g}{\partial x^2}=\frac{1}{y} \frac{\partial u_g}{\partial y}+\frac{\partial^2 u_g}{\partial y^2}\)
3
\(\frac{\partial^2 u_g}{\partial x^2}=\frac{2}{y} \frac{\partial u_g}{\partial y}-\frac{\partial^2 u_g}{\partial y^2}\)
4
\(\frac{\partial^2 u_g}{\partial x^2}=\frac{1}{y} \frac{\partial u_g}{\partial y}-\frac{\partial^2 u_g}{\partial y^2}\)