Final answer:
The general solution to the higher-order differential equation y(4) + y" + y' = 0 can be found by solving its characteristic equation and constructing the solution from its roots, which involves a combination of exponential and polynomial terms.
Step-by-step explanation:
To find the general solution of the given higher-order differential equation y(4) + y" + y' = 0, we apply methods from differential equations typically covered in college-level mathematics. First, we must find the characteristic equation associated with the differential equation. This is achieved by replacing each derivative with a power of a variable, usually 'r', which yields the characteristic equation r^4 + r^2 + r = 0. We can factor this into r(r^3 + r + 1) = 0. The root r = 0 is evident, and the cubic equation needs to be solved for the remaining roots.
Finding the roots of the cubic is a more complex process and may require numerical methods if the roots cannot be factored simply. Once the roots of the characteristic equation are determined, we use them to construct the general solution of the differential equation, which combines exponential and polynomial terms based on the nature of the roots (real, complex, repeated).
Without more specific information about the roots, we cannot provide the explicit form of the solution, but the general approach to solving such higher-order linear homogeneous differential equations remains consistent as described above.