Final answer:
The general solutions of the given differential equations involve determining the roots of their characteristic equations and using them to express the solution as a combination of exponential functions.
Step-by-step explanation:
The differential equations presented require us to find the function's general solution by solving higher-order linear homogeneous differential equations. Such equations often involve characteristic equations and their roots that lead to exponential solutions.
First Differential Equation
The general solution format for the first differential equation 9y''' + 12y'' + 4y' = 0 will involve finding the roots of its characteristic equation, which can typically be written as r^3 + ar^2 + br + c = 0, where a, b, and c are constants. The roots of the characteristic equation will then guide us in writing the general solution as a linear combination of exponential functions or their derivatives based on the multiplicity of the roots.
Second Differential Equation
Similarly, for the second differential equation y'''' - 8y'' + 16y = 0, we'll find the characteristic equation and its roots. The roots lead us to the general solution in terms of exponential functions, following the model e^(mx), where m are the roots of the characteristic equation.