Each differentiation of `e^(rt)` inserts a factor of `r`, so the second derivative is `r^2e^(rt).` This function is proportional to `e^(rt)` with proportionality constant `r^2`, which is always positive. The differential equation for motion of the spring-mass system has the second derivative of `x` proportional to `x` itself with proportionality constant `-k text[/]m,` which must be negative. So no exponential function can be a solution.