Question

Find the general solution to y'' + 6y' = 0. Provide the solution in the form y = ... and use c_1 and c_2 to denote arbitrary constants.

Answer 1

The general solution to the differential equation y'' + 6y' = 0 is y = c_1 + c_2e^{-6t}, where c_1 and c_2 are arbitrary constants.

Step-by-step explanation:

To find the general solution to the differential equation y'' + 6y' = 0, we first solve the characteristic equation associated with it. This involves finding the roots of the characteristic polynomial r^2 + 6r = 0. This polynomial factors as r(r + 6) = 0, giving us the roots r = 0 and r = -6.

The general solution to the differential equation combines solutions corresponding to each root. For r = 0, we have the solution y = c_1. For the root r = -6, we have the solution y = c_2e^{-6t}. Summing these solutions gives us the general solution to the differential equation:

y = c_1 + c_2e^{-6t}

where c_1 and c_2 are arbitrary constants determined by initial conditions.