Question:
\[
\frac{1}{D^{3}-3D^{2}+4}\,e^{2x}=\_
\]
\[
\frac{1}{D^{3}-3D^{2}+4}\,e^{2x}=\_
\]
Show Hint
The power of $x$ in the numerator matches the number of times you had to differentiate to get a non-zero denominator.
Updated On: Jul 4, 2026
- $\frac{e^{2x}}{6}$
- $xe^{2x}$
- $\frac{x^{2}e^{2x}}{6}$
- $\frac{xe^{-2x}}{6}$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
For $P.I. = \frac{1}{f(D)} e^{ax}$, we substitute $D = a$. If $f(a) = 0$, we multiply by $x$ and differentiate the denominator.
Step 2: Meaning
$f(D) = D^{3} - 3D^{2} + 4$. Substituting $D = 2$: $f(2) = 2^{3} - 3(2^{2}) + 4 = 8 - 12 + 4 = 0$. This is a case of failure.
Step 3: Analysis
Differentiate denominator: $f'(D) = 3D^{2} - 6D$. Substitute $D = 2$: $f'(2) = 3(4) - 6(2) = 12 - 12 = 0$. Failure again. Differentiate again: $f''(D) = 6D - 6$.
Step 4: Conclusion
Now substitute $D = 2$ in the second derivative: $f''(2) = 6(2) - 6 = 6$. The result is $\frac{x^{2}}{f''(2)} e^{2x} = \frac{x^{2}e^{2x}}{6}$. Final Answer: (C)
For $P.I. = \frac{1}{f(D)} e^{ax}$, we substitute $D = a$. If $f(a) = 0$, we multiply by $x$ and differentiate the denominator.
Step 2: Meaning
$f(D) = D^{3} - 3D^{2} + 4$. Substituting $D = 2$: $f(2) = 2^{3} - 3(2^{2}) + 4 = 8 - 12 + 4 = 0$. This is a case of failure.
Step 3: Analysis
Differentiate denominator: $f'(D) = 3D^{2} - 6D$. Substitute $D = 2$: $f'(2) = 3(4) - 6(2) = 12 - 12 = 0$. Failure again. Differentiate again: $f''(D) = 6D - 6$.
Step 4: Conclusion
Now substitute $D = 2$ in the second derivative: $f''(2) = 6(2) - 6 = 6$. The result is $\frac{x^{2}}{f''(2)} e^{2x} = \frac{x^{2}e^{2x}}{6}$. Final Answer: (C)
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