Question:
The complementary function of
\[
(D-2)^2y=8e^{-x}
\]
is
The complementary function of
\[
(D-2)^2y=8e^{-x}
\]
is
Show Hint
For a repeated auxiliary root \(m\), the complementary function is \((c_1+c_2x)e^{mx}\).
Updated On: Jul 4, 2026
- \((c_1+c_2x)e^x\)
- \((c_1+c_2x)e^{-x}\)
- \((c_1+c_2x)e^{-2x}\)
- \((c_1+c_2x)e^{2x}\)
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
The complementary function is obtained from the homogeneous part of the differential equation. Given, \[ (D-2)^2y=8e^{-x} \] For complementary function, we solve \[ (D-2)^2y=0 \]
Step 1: Write the auxiliary equation. Replace \(D\) by \(m\): \[ (m-2)^2=0 \]
Step 2: Find the roots. \[ (m-2)^2=0 \] \[ m=2,2 \] So \(m=2\) is a repeated root.
Step 3: Write the complementary function. For repeated root \(m\), the complementary function is \[ (c_1+c_2x)e^{mx} \] Here, \[ m=2 \] Therefore, \[ C.F.=(c_1+c_2x)e^{2x} \]
Step 4: Final answer. \[ \boxed{(c_1+c_2x)e^{2x}} \]
The complementary function is obtained from the homogeneous part of the differential equation. Given, \[ (D-2)^2y=8e^{-x} \] For complementary function, we solve \[ (D-2)^2y=0 \]
Step 1: Write the auxiliary equation. Replace \(D\) by \(m\): \[ (m-2)^2=0 \]
Step 2: Find the roots. \[ (m-2)^2=0 \] \[ m=2,2 \] So \(m=2\) is a repeated root.
Step 3: Write the complementary function. For repeated root \(m\), the complementary function is \[ (c_1+c_2x)e^{mx} \] Here, \[ m=2 \] Therefore, \[ C.F.=(c_1+c_2x)e^{2x} \]
Step 4: Final answer. \[ \boxed{(c_1+c_2x)e^{2x}} \]
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