Simplify your answer to the previous part and enter a differential equation in terms of the dependent variable xx satisfied by x=e5t. enter the derivatives of xx using prime notation (x′,x′′,x′′′x′,x″,x‴).

x'-5x=0, or x''-25x=0, or x'''-125x=0

Step-by-step explanation:

The function is infinitely differentiable, so it satisfies a infinite number of differential equations. The required answer depends on your previous part, so I will describe a general procedure to obtain the equations.

Using rules of differentiation, we obtain that . Differentiate again to obtain, . Repeating this process, .

This can repeated infinitely, so it is possible to obtain a differential equation of order n. The key is to differentiate the required number of times and write the equation in terms of x.