Final answer:
The general solution to the second-order linear homogeneous differential equation with constant coefficients, 9y'' + 9y' + 4y = 0, involves first finding the roots of the characteristic equation 9r^2 + 9r + 4 = 0, which dictate the form of the solution.
Step-by-step explanation:
The differential equation you're tasked with solving is a second-order linear homogeneous differential equation with constant coefficients. The given equation is 9y'' + 9y' + 4y = 0. To solve it, we look for solutions of the form y = ert, where r will be determined by the characteristic equation derived from the differential equation.
Substituting y = ert into the differential equation results in the characteristic equation: 9r2 + 9r + 4 = 0. Solving this quadratic equation for r using the quadratic formula yields the roots r, which dictates the form of the general solution. If the roots are real and distinct, the solution will consist of the sum of exponentials; if they are complex, the solution involves exponentials and trigonometric functions; and if they are repeated roots, the solution involves exponentials and polynomial terms.
In the case of complex solutions r = a ± bi, the general solution of the differential equation is y(t) = eat(A cos(bt) + B sin(bt)), where A and B are constants that can be determined by initial conditions.