Final answer:
To solve the differential equation y'' + 9y = 2sec(3x), we must find the complementary solution and the particular solution, then add them to find the general solution, which is then verified.
Step-by-step explanation:
Solving the Differential Equation y'' + 9y = 2sec(3x)
To solve the differential equation y'' + 9y = 2sec(3x), we first observe that this is a second-order linear non-homogeneous differential equation. The particular solution can be found using the method of undetermined coefficients, or the variation of parameters, after finding the complementary solution to the associated homogeneous equation y'' + 9y = 0. The complementary solution is y_c = C_1 cos(3x) + C_2 sin(3x), where C_1 and C_2 are constants determined by initial conditions.
To find the particular solution, denote it as y_p, we can guess a solution involving a function of x times sec(3x) and then differentiate and substitute back into the original equation to solve for the coefficients. Once y_p is found, the general solution to the differential equation is y = y_c + y_p.
Finally, a solution verification is performed by plugging the solution into the original differential equation to ensure that both sides are equal. If so, then the derived function is indeed the solution to the given differential equation.