For the differential equation $x\frac{dy}{dx}+2y= x^2\log_ex$
(A) Integrating factor is $2x$
(B) Integrating factor is $x^2$
(C) General Solution is $y=\frac{x^2}{16}(4 \log_e|x|-1)+Cx^{-2}$ Where C is an arbitrary constant.
(D) General Solution is $y=\frac{x^2}{16}(4 \log_e|x|-1)+C$ Where C is an arbitrary constant.
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (C) only
General linear differential equation:
$\frac{dy}{dx} + P(x)y = Q(x)$
Integrating Factor (IF):
$IF = e^{\int P(x)\,dx}$
General solution:
$y \cdot IF = \int Q(x) \cdot IF \, dx + C$
Given equation:
$\frac{dy}{dx} + \frac{2}{x}y = x\log_e|x|$
$IF = e^{\int \frac{2}{x}dx} = x^2$
$\Rightarrow \frac{d}{dx}(x^2y) = x^2 \cdot x\log_e|x| = x^3\log_e|x|$
$\int x^3\log_e|x|\,dx = \frac{x^4}{4}\log_e|x| - \frac{x^4}{16} + C$
$x^2y = \frac{x^4}{4}\log_e|x| - \frac{x^4}{16} + C$
$\Rightarrow y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$
Correct options: (B) and (C)