Question

Complementary function of the differential equation \(\dfrac{d^2x}{dt^2}+4x=a\sin t\cos t\) is

• A. \(x=e^t(c_1\cos t+c_2\sin t)\)
• B. \(x=c_1\cos 2t+c_2\sin 2t\)
• C. \(t=e^x(c_1\cos 2x+c_2\sin 2x)\)
• D. \(x=c_1e^{2t}+c_2e^{-2t}\)

Hint:

For auxiliary roots \(\pm bi\), the complementary function is \(c_1\cos bx+c_2\sin bx\).

Answer 1

The Correct Option is B

Concept:
The complementary function is obtained by solving the homogeneous part of the differential equation. Given, \[ \frac{d^2x}{dt^2}+4x=a\sin t\cos t \] For complementary function, take right hand side as zero: \[ \frac{d^2x}{dt^2}+4x=0 \]

Step 1: Form the auxiliary equation. Let \[ x=e^{mt} \] Then the auxiliary equation is \[ m^2+4=0 \]

Step 2: Solve for \(m\). \[ m^2=-4 \] \[ m=\pm 2i \]

Step 3: Write the complementary function. For roots \[ m=\pm bi \] the complementary function is \[ c_1\cos bt+c_2\sin bt \] Here, \[ b=2 \] Therefore, \[ x=c_1\cos 2t+c_2\sin 2t \]

Step 4: Final answer. \[ \boxed{x=c_1\cos 2t+c_2\sin 2t} \]