Question

Find the general solution to the homogeneous differential equation (d²y/dt² + 4y = 0). Use (c_1) and (c_2) to denote arbitrary constants.

Answer 1

The general solution to the homogeneous differential equation d²y/dt² + 4y = 0 is y(t) = c_1 cos(2t) + c_2 sin(2t), where c_1 and c_2 are arbitrary constants.

Step-by-step explanation:

The differential equation d²y/dt² + 4y = 0 is a second-order linear homogeneous differential equation with constant coefficients. To find the general solution, we can assume a solution of the form y = ert, where r is a constant that we need to determine. Substituting y into the differential equation gives us a characteristic equation r² + 4 = 0.

Solving the characteristic equation r² + 4 = 0, we find the roots to be r = ±2i. Therefore, the general solution to the differential equation is a linear combination of the functions cosine and sine. Specifically, the general solution can be written as y(t) = c_1 cos(2t) + c_2 sin(2t), where c_1 and c_2 are arbitrary constants.