Step 1: Conceptual Foundation:
The general solution to a linear, non-homogeneous, second-order differential equation is derived by summing the complementary function (\(y_c\)), which resolves the associated homogeneous equation, and a particular integral (\(y_p\)), which represents any valid solution to the non-homogeneous equation.
Step 2: Methodology Outline:
1. Determine the Complementary Function (\(y_c\)): Solve the auxiliary equation corresponding to \(y'' + 9y = 0\).2. Determine the Particular Integral (\(y_p\)): Employ either the method of undetermined coefficients or the operator method. Due to the forcing term \(\cos(3x)\) being present in the complementary function, this scenario is classified as resonance.
Step 3: Detailed Derivation:
1. Derivation of \(y_c\):
The governing homogeneous equation is \(y'' + 9y = 0\).The resultant auxiliary equation is \(m^2 + 9 = 0 \implies m^2 = -9 \implies m = \pm 3i\).The derived complementary function is \(y_c = C_1\cos(3x) + C_2\sin(3x)\).
2. Derivation of \(y_p\):
The forcing function is \(\cos(3x)\), which is observed within \(y_c\). This situation indicates resonance.The operator method is applied for the particular integral calculation:\[ y_p = \frac{1}{D^2+9} \cos(3x) \]As direct substitution of \(D^2 = -a^2 = -3^2 = -9\) results in a zero denominator, the resonance formula is invoked:\[ \frac{1}{D^2+a^2} \cos(ax) = \frac{x}{2a} \sin(ax) \]With \(a=3\).\[ y_p = \frac{x}{2(3)} \sin(3x) = \frac{x}{6}\sin(3x) \]3. Assembly of the General Solution:
The comprehensive solution is expressed as \(y = y_c + y_p\).\[ y(x) = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \]Step 4: Conclusive Result:
The general solution is determined to be \( y(x) = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \), aligning with option (B).