Question

We have the differential equation y" - 4y = 0. The first solution is given as y1 = cos(2x). We are instructed to find a second solution y2(x).

Answer 1

The second solution to the differential equation y" - 4y = 0, given the first solution y1 = cos(2x), is y2 = sin(2x), resulting in the general solution y = A cos(2x) + B sin(2x).

Step-by-step explanation:

The differential equation given is y" - 4y = 0. Knowing that one solution is y1 = cos(2x), we seek another linearly independent solution y2(x) to form the general solution. By employing the method of reduction of order, or recognizing that the characteristic equation of the differential equation suggests solutions of the form erx, we determine that the missing solution involves hyperbolic functions or the sine function at the same frequency due to the nature of the characteristic roots (real and repeated). Thus, a second solution could be y2 = sin(2x).

The full general solution to the differential equation is therefore y = A cos(2x) + B sin(2x), where A and B are determined by initial conditions or additional constraints.