(A) Given differential equation: $x \cos y \,dy = (x e^x \log x + e^x) dx$ Divide both sides by $x$ (assuming $x \neq 0$): $\cos y \,dy = \left(e^x \l

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(A) Given differential equation: $x \cos y \,dy = (x e^x \log x + e^x) dx$ Divide both sides by $x$ (assuming $x \neq 0$): $\cos y \,dy = \left(e^x \log x + \frac{e^x}{x}\right) dx$ Integrating both sides: $\int \cos y \,dy = \int e^x \left(\log x + \frac{1}{x}\right) dx$ Using the standard integral formula $\int e^x \{f(x) + f'(x)\} dx = e^x f(x) + C$, where $f(x) = \log x$ and $f'(x) = \frac{1}{x}$: $\sin y = e^x \log x + C$

Class 12 Mathematics Differential Equations Variable separable type differential equations

Medium

General solution of the differential equation $x \cos y \,dy = (x e^x \log x + e^x) dx$ is (where $C$ is a constant of integration.)

MHT CET 2022 All MHT CET

A
$\sin y = e^x \log x + C$
B
$\sin y = e^x + C \log x$
C
$\sin y = C e^x + \log x$
D
$e^x \sin y = \log x + C$

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Differential Equations

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Variable separable type differential equations

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General solution of the differential equation $x \cos y \,dy = (x e^x \log x + e^x) dx$ is (where $C$ is a constant of integration.)

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