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What is the solution of the following differential equation?

\(\rm \ln\left(\frac{dy}{dx}\right)+y = x\)


1. ex + ey = c
2. ex + y = c
3. ex - ey = c
4. ex - y = c

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Correct Answer - Option 3 : ex - ey = c

Formula Used:

If lnx = n, then x = en

ex dx = ex + c

Calculation:

\(\rm \ln\left(\frac{dy}{dx}\right)+y= x\)

⇒ \(\rm \ln\left(\frac{dy}{dx} \right)\) = x - y

⇒ \(\rm \frac{dy}{dx}\) = ex - y = \(\rm \frac{e^x}{e^y}\), which is in variable separable form.

⇒ ex dx = ey dy

Integrating both sides, we get:

⇒ ex = ey + c

∴ The solution of the differential equation is ex - ey = c.

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