Question

The complementary function of \[ (D-2)^2y=8e^{-x} \] is

• A. \((c_1+c_2x)e^x\)
• B. \((c_1+c_2x)e^{-x}\)
• C. \((c_1+c_2x)e^{-2x}\)
• D. \((c_1+c_2x)e^{2x}\)

Hint:

For a repeated auxiliary root \(m\), the complementary function is \((c_1+c_2x)e^{mx}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The complementary function (C.F.) is the general solution to the associated homogeneous differential equation. To find the C.F., we set the right-hand side of the non-homogeneous equation to zero and solve the resulting homogeneous equation.

Step 2: Key Formula or Approach:
1. Consider the associated homogeneous equation: \((D - 2)^2y = 0\). 2. Write down the auxiliary equation by replacing \(D\) with \(m\): \((m - 2)^2 = 0\). 3. Find the roots of the auxiliary equation. 4. Use the roots to write the complementary function. For repeated real roots \(m_1 = m_2 = m\), the solution is \(y_c = (c_1 + c_2x)e^{mx}\).

Step 3: Detailed Explanation:
The given differential equation is: \[ (D - 2)^2y = 8e^x \] To find the complementary function, we solve the associated homogeneous equation: \[ (D - 2)^2y = 0 \] The auxiliary equation is: \[ (m - 2)^2 = 0 \] This equation has a repeated real root: \[ m = 2, 2 \] When the auxiliary equation has a repeated real root \(m\), the complementary function is of the form \(y_c = (c_1 + c_2x)e^{mx}\). In our case, \(m=2\), so the complementary function is: \[ y_c = (c_1 + c_2x)e^{2x} \]

Step 4: Final Answer:
The complementary function for the given differential equation is \((c_1 + c_2x)e^{2x}\), which matches option (D).