Final answer:
To find y as a function of x, we need to solve the characteristic equation of the given third-order linear homogeneous differential equation and use the initial conditions to find the coefficients of the solution.
Step-by-step explanation:
We are given the third-order linear homogeneous differential equation y''' - 9y'' - y' + 9y = 0 with initial conditions y(0) = 0, y'(0) = 9, and y''(0) = 0. To solve this, we first find the characteristic equation which is r^3 - 9r^2 - r + 9 = 0. This characteristic equation can be solved for r to find the roots which represent the solution to the differential equation.
Once the roots are found, the general solution to the differential equation is a linear combination of exponential functions based on these roots. The coefficients of these exponential functions are then determined by using the given initial conditions. In this case, since we don't have an explicit equation to solve for the roots, a more detailed calculation would be needed to find the coefficients that satisfy the initial conditions.
After determining the coefficients, we will have the specific solution y(x) which defines the dependent variable y as a function of the independent variable x, complying with the given initial conditions.