Question

The general solution of differential equation \( \frac{d^2y}{dx^2} + 9y = \cos(3x) \) is:

• A. \( C_1\cos(3x) + C_2\sin(3x) + \frac{x}{3}\sin(3x) \)
• B. \( C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \)
• C. \( C_1\cos(3x) + C_2\sin(3x) - \frac{x}{6}\sin(3x) \)
• D. \( C_1\cos(3x) + C_2\sin(3x) + \frac{x}{3}\cos(3x) \)

Hint:

When solving \( (D^2+a^2)y = \cos(ax) \) or \( (D^2+a^2)y = \sin(ax) \), always check if the roots of the auxiliary equation (\(\pm ai\)) match the frequency of the forcing term. If they do, you are in a resonance case, and the particular solution will involve multiplication by x. Remember the formulas:
\( \frac{1}{D^2+a^2}\cos(ax) = \frac{x}{2a}\sin(ax) \)
\( \frac{1}{D^2+a^2}\sin(ax) = -\frac{x}{2a}\cos(ax) \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The general solution of a linear non-homogeneous second-order differential equation is the sum of the complementary function (\(y_c\)), which is the solution to the homogeneous equation, and a particular integral (\(y_p\)), which is any solution to the non-homogeneous equation.
Step 2: Key Formula or Approach:
1. Find the Complementary Function (\(y_c\)): Solve the auxiliary equation of \(y'' + 9y = 0\). 2. Find the Particular Integral (\(y_p\)): Use the method of undetermined coefficients or the operator method. Since the forcing term \(\cos(3x)\) is part of the complementary function, this is a case of resonance.
Step 3: Detailed Explanation:
1. Find \(y_c\):
The homogeneous equation is \(y'' + 9y = 0\). The auxiliary equation is \(m^2 + 9 = 0 \implies m^2 = -9 \implies m = \pm 3i\). The complementary function is \(y_c = C_1\cos(3x) + C_2\sin(3x)\).
2. Find \(y_p\):
The forcing term is \(\cos(3x)\), which appears in \(y_c\). This is the case of resonance. We use the operator method for the particular integral: \[ y_p = \frac{1}{D^2+9} \cos(3x) \] Since substituting \(D^2 = -a^2 = -3^2 = -9\) makes the denominator zero, we use the resonance formula: \[ \frac{1}{D^2+a^2} \cos(ax) = \frac{x}{2a} \sin(ax) \] Here, \(a=3\). \[ y_p = \frac{x}{2(3)} \sin(3x) = \frac{x}{6}\sin(3x) \] 3. Combine to form the general solution:
The general solution is \(y = y_c + y_p\). \[ y(x) = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \] Step 4: Final Answer:
The general solution is \( y(x) = C_1\cos(3x) + C_2\sin(3x) + \frac{x}{6}\sin(3x) \), which matches option (B).