Match List-I with List-II
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List-I Differential Equation |
List-II Integrating factor |
|
(A) $x\frac{dy}{dx}-y = 2x^2$ |
(I) $e^{-y}$ |
|
(B) $\frac{dy}{dx}+\frac{y}{x}=2x$ |
(II) $\frac{1}{x}$ |
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(C) $x\frac{dy}{dx}+ 2y = x^2\log x$ |
(III) $x$ |
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(D) $\frac{dx}{dy}-x=y$ |
(IV) $x^2$ |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
|
List-I Differential Equation |
List-II Integrating factor |
|
(A) $x\frac{dy}{dx}-y = 2x^2$ |
(II) $\frac{1}{x}$ |
|
(B) $\frac{dy}{dx}+\frac{y}{x}=2x$ |
(III) $x$ |
|
(C) $x\frac{dy}{dx}+ 2y = x^2\log x$ |
(IV) $x^2$ |
|
(D) $\frac{dx}{dy}-x=y$ |
(I) $e^{-y}$ |
Each equation is reduced to linear form and integrating factor identified.
(A) $x\frac{dy}{dx}-y=2x^2$
$\frac{dy}{dx}-\frac{1}{x}y=2x$
Integrating factor $=e^{\int -\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$
(A) → (II)
(B) $\frac{dy}{dx}+\frac{y}{x}=2x$
Integrating factor $=e^{\int \frac{1}{x}dx}=x$
(B) → (III)
(C) $x\frac{dy}{dx}+2y=x^2\log x$
$\frac{dy}{dx}+\frac{2}{x}y=x\log x$
Integrating factor $=e^{\int \frac{2}{x}dx}=e^{2\log x}=x^2$
(C) → (IV)
(D) $\frac{dx}{dy}-x=y$
$\frac{dx}{dy}-x=y$ is linear in $x$ with variable $y$
Integrating factor $=e^{\int (-1)dy}=e^{-y}$
(D) → (I)
Final Matching: (A)-(II), (B)-(III), (C)-(IV), (D)-(I).