Question

The complementary function of the differential equation \( y'' + 16y = x \sin px, \, p = 4, \) is given by:

• A. \( A \cos(16x) + B \sin(16x), \) where \( A \) and \( B \) are constants.
• B. \( A \cos(4x) + B \sin(4x), \) where \( A \) and \( B \) are constants.
• C. \( A \cos(8x) + B \sin(8x), \) where \( A \) and \( B \) are constants.
• D. \( A \cos(2x) + B \sin(2x), \) where \( A \) and \( B \) are constants.

Hint:

For constant-coefficient differential equations, use the characteristic equation to find the complementary function.

Answer 1

The Correct Option is B

The complementary function solves the homogeneous equation \( y'' + 16y = 0 \). Here, the characteristic equation is \( r^2 + 16 = 0 \), giving roots \( \pm 4i \). The solution is:
\( y_c = A \cos(4x) + B \sin(4x). \)