Problem $4.$ Assume the variable $L$ is defined as:

$L=\frac{1}{2}m{x}^{}(t{)}^2-\frac{1}{2}kx(t{)}^2$

where $m$ and $k$ are constants. Apply the following equation

$\frac{d}{dt}(\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}(x^{})})-\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}\text{x}}=0$

to derive the differential equation. Then, for $k=m=1$ with the initial conditions $x(0)=2$ and $x^{}(0)=0,$ solve for $x(t).$