Question

Assertion (A) : One of the particular solutions of the differential equation \(\frac{dy}{dx} = e^{x+y}\) can be \(e^x + e^{-y} = -2\).
Reason (R) : \(e^x + e^{-y} = C\) is the general solution of the differential equation \(\frac{dy}{dx} = e^{x+y}\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Always check the range of functions in Assertion-Reason questions. Exponential and squared terms are always non-negative. Even if the algebraic form of a solution matches, it must be mathematically possible within the range of the functions involved.

Answer 1

The Correct Option is D

Concept:
• To find the general solution, use the method of variable separable.
• Note that \(e^{x+y} = e^x \cdot e^y\).
• The domain of exponential functions \(e^u\) is always positive (\(e^u > 0\) for all real \(u\)).

Step 1: Solve the differential equation
Given: \[ \frac{dy}{dx} = e^x \cdot e^y \] Separate the variables: \[ e^{-y} \, dy = e^x \, dx \] Integrate both sides: \[ \int e^{-y} \, dy = \int e^x \, dx \] \[ -e^{-y} = e^x + C' \] \[ e^x + e^{-y} = -C' \] Let \(-C' = C\), so the general solution is \(e^x + e^{-y} = C\). Thus, Reason (R) is true.

Step 2: Analyze the Assertion
The Assertion states that a particular solution can be \(e^x + e^{-y} = -2\). For any real \(x\) and \(y\), \(e^x > 0\) and \(e^{-y} > 0\). The sum of two positive numbers must be positive: \[ e^x + e^{-y} > 0 \] Since \(e^x + e^{-y}\) can never be \(-2\), the Assertion (A) is false.