Final answer:
To find y as a function of t, solve the characteristic equation for the given second-order linear homogeneous differential equation. Determine the roots to form the general solution and apply the initial conditions to find the specific solution for y(t).
Step-by-step explanation:
The question involves finding y as a function of t for a second-order linear homogeneous differential equation with constant coefficients. The differential equation given is y'' + 18y' + 82y = 0, and initial conditions are y(0) = 2 and y'(0) = 9.
To solve this equation, we first find the characteristic equation which is r^2 + 18r + 82 = 0. We solve for r to find the roots of the characteristic equation, which will determine the form of the solution. If the roots are real and distinct, the general solution will be of the form y(t) = c1e^(r1t) + c2e^(r2t). If the roots are real and repeated, the general solution will be of the form y(t) = (c1 + c2t)e^(rt). However, if the roots are complex, the solution will be y(t) = e^(at)(c1 cos(bt) + c2 sin(bt)), where a is the real part and b is the imaginary part of the complex roots. Finally, we use the initial conditions to solve for constants c1 and c2.