The solution of the differential equation d y/d x+y y x=0 is

Correct Answer is : sin⁡xlog⁡y=c x y=csinxlogy=c

Correct Answer is : sin⁡xlog⁡y=c x y=csinxlogy=c

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COMEDK 2025 Morning Solved Paper

Question : 166 of 180

Marks: +1, -0

\frac{d y}{d x}+y \log y \cot x=0

Solution:

Concept:

Separate the variables then integrate to solve the differential equation.

Explanation:

Given

\frac{dy}{dx} + y \log y \cot x = 0

Rearrange:

\frac{dy}{dx} = -y \log y \cot x

Separate variables:

\frac{dy}{y \log y} = -\cot x \, dx

Integrate both sides.

Left side: let

u = \log y

, then

du = \frac{dy}{y}

, so

\int \frac{dy}{y \log y} = \int \frac{du}{u} = \ln |\log y|

Right side:

\int -\cot x \, dx = -\ln |\sin x|

Equate:

\ln |\log y| = -\ln |\sin x| + C

Combine logs:

\ln |\log y| + \ln |\sin x| = C

\Rightarrow \ln | (\log y)(\sin x) | = C

Exponentiate:

(\log y)(\sin x) = e^C = \text{constant}

Thus the general solution is

\sin x \, \log y = c

, where

c

is an arbitrary constant.

Answer:

Option C:

\sin x \, \log y = c

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