Final answer:
To solve the differential equation y''+36y=12sin(6x), we use the method of undetermined coefficients. The solution includes a homogeneous part, yh = C1 cos(6x) + C2 sin(6x), and a particular solution yp = Ax sin(6x), where A is found by substituting yp back into the original equation.
Step-by-step explanation:
The task is to solve the differential equation y''+36y=12sin(6x) using the method of undetermined coefficients. First, we solve the homogeneous equation y''+36y=0, which has the solution yh = C1 cos(6x) + C2 sin(6x). Next, for the non-homogeneous part, we guess a particular solution of the form yp = A sin(6x) since the right-hand side is a sine function of 6x.
Substituting yp into the differential equation gives -36A sin(6x) + 36A sin(6x) = 12sin(6x), which simplifies to 0 = 12sin(6x), indicating a discrepancy because the choice of A sin(6x) would yield a zero on the left-hand side. Thus, the correct guess should account for this discrepancy. Considering that the differential operator L[y]=y''+36y annihilates both sine and cosine functions of 6x, we multiply our guess by x to obtain yp = Ax sin(6x).
Plugging in yp into the original differential equation, we need to differentiate twice and find the coefficient A such that the equation is satisfied. Finally, the complete solution is the sum of the homogeneous and particular solutions: y(x) = C1 cos(6x) + C2 sin(6x) + Ax sin(6x), where A is determined from substituting back into the differential equation and C1, C2 are determined from initial conditions if given.