Final answer:
To find y as a function of x for the given differential equation, determine the characteristic equation's roots and construct the general solution based on those roots.
Step-by-step explanation:
The problem requires us to find y as a function of x given the third-order linear homogeneous differential equation y''' - 12y' + 35y' = 0.
To solve this differential equation, we first need to find the characteristic equation which is of the form r^3 - 12r + 35r = 0. This simplifies to r(r^2 - 12 + 35) = 0, giving us the roots for r. These roots are the key to finding the general solution to the differential equation.
Once we have the roots, we can use them to construct the general solution. Depending on whether the roots are real or complex, and whether they are distinct or repeated, the form of the solution will vary. In this case, assume the roots are real and distinct, and so the general solution is of the form y = C1 * e^(r1x) + C2 * e^(r2x) + C3 * e^(r3x), where C1, C2, and C3 are constants determined by initial conditions or boundary values