Question

This problem is related to Problem 7.5 in the text. Consider the differential equation d^2v(t) / dt^2 + 9 dv(t)/ dt + 14v(t) = 0

Which of the following functions are solutions to the differential equation?

A. C1e (2+7)t О

В. Сје2 C. C1e 2 + C2e_7t D. Cie7t E. C1e2 C2 F.Cie -7t + C2 G. C1e 2t + C2 О

Н. Се 7t I.Cie 2t J. All of the above

K. None of the above

Answer 1

The characteristic equation for the given differential equation is:

r^2 + 9r + 14 = 0

Factoring the equation, we get:

(r + 7)(r + 2) = 0

Therefore, the roots of the characteristic equation are -7 and -2.

The general solution to the differential equation is then:

v(t) = C1e^(-7t) + C2e^(-2t)

Therefore, options A, B, D, F, G, H, and J are not solutions to the differential equation.

Option C can be simplified as:

v(t) = C1e^(-7t) + C2e^(-2t)

Therefore, option C is a solution to the differential equation.

Option E can be simplified as:

v(t) = (C1 + C2t)e^(-2t)

Therefore, option E is not a solution to the differential equation.

Option I can be simplified as:

v(t) = C1cos(2t) + C2sin(2t)

Therefore, option I is not a solution to the differential equation.

Therefore, the correct answer is K. None of the above.