Question:
The particular integral of
\[
\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=e^{-2x}
\]
is
The particular integral of
\[
\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=e^{-2x}
\]
is
Show Hint
If \(F(a)=0\) for RHS \(e^{ax}\), use the repeated/root adjustment method and multiply by \(x\).
Updated On: May 7, 2026
- \(-xe^{-2x}\)
- \(xe^{-2x}\)
- \(-\frac{x}{2}e^{-2x}\)
- \(\frac{x}{2}e^{-2x}\)
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Given: \[ (D^2+3D+2)y=e^{-2x} \]
Step 2: Factor operator: \[ D^2+3D+2=(D+1)(D+2) \]
Step 3: For RHS \(e^{-2x}\), put \(D=-2\): \[ F(-2)=(-2)^2+3(-2)+2 \] \[ =4-6+2=0 \]
Step 4: Since \(F(-2)=0\), ordinary substitution fails. Because \((D+2)\) is a factor, multiply by \(x\).
Step 5: The particular integral becomes: \[ PI=-xe^{-2x} \] \[ \boxed{-xe^{-2x}} \]
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