Question

Find the general solution to the homogeneous differential equation dt2 / d2y + 6 dt/dy +9y =0The solution can be written in the form y=C1f1(t) + C2f2(t) with f1(t) = _____ and f2(t) _____. Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that the functions are normalized with f1(0)=1 and f2(0)=0

Answer 1

The general solution to the homogeneous differential equation is y(t) = C1e^-3t + C2te^-3t. The functions normalized with f1(0)=1 and f2(0)=0 are f1(t) = 1 and f2(t) = te^-3t.

Step-by-step explanation:

The given homogeneous second-order linear differential equation is dt^2 / d^2y + 6 dt/dy + 9y = 0. To find the general solution, we assume a solution of the form y = ert. Substituting into the differential equation, we get the characteristic equation r^2 + 6r + 9 = 0, which can be factored as (r + 3)2 = 0. The repeated root is r = -3.

For repeated roots, the general solution can be expressed as y(t) = C1e-^3t + C2te^-3t, where f1(t) = e^-3t and f2(t) = te^-3t. Normalizing f1(t) and f2(t) gives f1(t) = e^-3t/e0 = 1 when t = 0 and f2(t) = t * e^-3t/e0 = 0 when t = 0.