$(5$ points$)$ Consider the differential equation $f^{\text{'}\text{'}}(t)f(t)=e^t.$ Show explicitly for arbitrary constants $c_1$ and $c_2$ that $f_1(t)=c_{1}cos(t)\frac{1}{2}{e}^t$ and $f_2(t)=c_{2}sin(t)\frac{1}{2}{e}^t$ are solutions to this equation but $f_1(t)f_2(t)$ is not. This is due to the inhomogeneity in the system. Hint: plug in$,$ do not try to solve the differential equation.