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.