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Sliva [168]
2 years ago
5

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‴).
Mathematics
1 answer:
rusak2 [61]2 years ago
7 0

Answer:

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

Step-by-step explanation:

The function x(t)=e^{5t} 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 x'(t)=5e^{5t}=5x \text{ then }x'-5x=0. Differentiate again to obtain, x''(t)=25e^{5t}=25x=5x' \text{ then }x''-25x=0=x''-5x'. Repeating this process, x'''(t)=125e^{5t}=125x=25x' \text{ then }x'''-125x=0=x'''-25x'.

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.

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