Let c be a positive real number and let u: R2 → R be defined by \(u(x, t)=\frac{1}{2 c} \int_{x-c t}^{x+c t} e^{s^{2}} d s \text { for }(x, t) \in \mathbb{R}^{2}\). Then which one of the following is true?
1
\(\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}} \text { on } \mathbb{R}^{2}\)
2
\(\frac{\partial u}{\partial t}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}\text { on } \mathbb{R}^{2}\)
3
\(\frac{\partial u}{\partial t} \frac{\partial u}{\partial x}=0 \text { on } \mathbb{R}^{2}\)
4
\(\frac{\partial^{2} u}{\partial t \partial x}=0 \text { on } \mathbb{R}^{2}\)