Final answer:
The question is about solving a non-homogeneous differential equation on an interval. The key aspect is finding a particular solution for the non-homogeneous term and a general solution for the homogeneous counterpart. The additional information provided in the question does not have a direct relation to the solution process.
Step-by-step explanation:
The student has asked a question regarding the solution of a non-homogeneous differential equation y'' + 9y = sec²(3x) on a given interval. Solving such an equation typically involves finding a particular solution that fits the non-homogeneous part (sec²(3x)) and a general solution to the corresponding homogeneous equation (y'' + 9y = 0). The solution to the homogeneous equation is a combination of sine and cosine functions, while the particular solution often requires a method such as undetermined coefficients or variation of parameters.
However, based on the information provided, it is unclear how the listed unknowns (y1, y2, y3; V1, V2, V3), equations, and other functions provided in the question are relevant to solving the differential equation at hand. To properly solve this differential equation, we would focus solely on the techniques for solving non-homogeneous differential equations, without the extraneous details provided.