The solution of the differential equation: x y d y=≤ft(x e^x x+e^x) d x is

Correct Answer is : sin⁡y=exlog⁡x+c y=e^x x+csiny=exlogx+c

Correct Answer is : sin⁡y=exlog⁡x+c y=e^x x+csiny=exlogx+c

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COMEDK 2025 Evening Solved Paper

Question : 143 of 180

Marks: +1, -0

x \cos y d y=\left(x e^x \log x+e^x\right) d x

Solution:

Concept:

Separate variables and integrate after recognizing a derivative pattern.

Explanation:

Rewrite the equation as

\cos y \, dy = e^x\left(\ln x + \frac{1}{x}\right) dx

Notice that

\frac{d}{dx}(e^x \ln x) = e^x \ln x + e^x \cdot \frac{1}{x} = e^x\left(\ln x + \frac{1}{x}\right)

Integrate both sides:

\int \cos y \, dy = \int e^x\left(\ln x + \frac{1}{x}\right) dx

The left side gives

\sin y

, and the right side gives

e^x \ln x + C

Thus the general solution is

\sin y = e^x \ln x + C

Answer:

\sin y = e^x \log x + c

, which corresponds to option B.

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