Question

For Particular Integral, Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	\( \frac{1}{(D-1)} x^2 \)	I.	\( xe^x \)
B.	\( \frac{1}{D^2+D+1} \cos x \)	II.	\( \sin x \)
C.	\( \frac{1}{(D-1)^2} e^x \)	III.	\( \frac{x^2 e^x}{2} \)
D.	\( \frac{1}{D^3-3D^2+4D-2} e^x \)	IV.	\( -(x^2 + 2x + 2) \)

(Note: List-I Item A is assumed to be \( \frac{1}{D-1} x^2 \) based on the options)

• A. A - I, B - II, C - III, D - IV
• B. A - I, B - III, C - II, D - IV
• C. A - IV, B - II, C - III, D - I
• D. A - IV, B - II, C - I, D - III

Hint:

Master the different methods for finding particular integrals based on the form of the function on the right side: exponential (\(e^{ax}\)), trigonometric (\(\sin(ax), \cos(ax)\)), polynomial (\(x^n\)), and their products. Special attention is needed for "cases of failure" where the simple substitution method fails.

Answer 1

The Correct Option is C

Step 1: Solve for the particular integral of each expression. - B. \(\frac{1}{D^2+D+1}\cos x\): For cosine functions, we substitute \(D^2 \to -a^2\), so \(D^2 \to -1^2 = -1\). \[ P.I. = \frac{1}{-1+D+1}\cos x = \frac{1}{D}\cos x = \int \cos x \,dx = \sin x \] This matches II. - C. \(\frac{1}{(D-1)^2e^x\):} This is a case of failure, as substituting \(D=1\) makes the denominator zero. We use the formula \(\frac{1}{f(D)}e^{ax}V = e^{ax}\frac{1}{f(D+a)}V\). \[ P.I. = e^x \frac{1}{((D+1)-1)^2}(1) = e^x \frac{1}{D^2}(1) = e^x \iint 1 \,dx\,dx = e^x \frac{x^2}{2} \] This matches III. - D. \(\frac{1}{D^3-3D^2+4D-2}e^x\): Substitute \(D=1\): \(1-3+4-2=0\). It's a failure case. We use the rule: If \(f(a)=0\), P.I. is \(x \frac{1}{f'(a)}e^{ax}\). Let \(f(D) = D^3-3D^2+4D-2\). Then \(f'(D) = 3D^2-6D+4\). \(f'(1) = 3(1)^2 - 6(1) + 4 = 3-6+4=1\). \[ P.I. = x \frac{1}{1}e^x = xe^x \] This matches I. - A. \(\frac{1}{(D-1)x^2\):} We use binomial expansion: \(\frac{1}{D-1} = -(1-D)^{-1} = -(1+D+D^2+...)\). \[ P.I. = -(1+D+D^2)(x^2) = -(x^2 + D(x^2) + D^2(x^2)) = -(x^2 + 2x + 2) \] This matches IV.
Step 2: Formulate the correct matching sequence. The matches are: A\(\rightarrow\)IV, B\(\rightarrow\)II, C\(\rightarrow\)III, D\(\rightarrow\)I. This corresponds to option (3).