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 separate indices first when exponential functions contain sums: \( e^{x+y} \to e^x e^y \). This makes it easy to spot that the equation is solvable via direct variable separation.

Answer 1

The Correct Option is A

Concept: To find the solution of the given differential equation, we use the variable-separable method. A general solution contains an arbitrary constant \( C \), while a particular solution is obtained by assigning a specific numerical value to \( C \).

Step 1: Solve the differential equation by separating variables.
The given equation is: \[ \frac{dy}{dx} = e^{x+y} = e^x \cdot e^y \] Separating the variables \( x \) and \( y \) onto opposite sides: \[ \frac{1}{e^y} \, dy = e^x \, dx \quad \Rightarrow \quad e^{-y} \, dy = e^x \, dx \]

Step 2: Integrate both sides.
Integrating both sides of the separated equation: \[ \int e^{-y} \, dy = \int e^x \, dx \] \[ -e^{-y} = e^x + C_1 \] Rearranging the terms to align with the standard equation forms: \[ e^x + e^{-y} = -C_1 \] Let \( -C_1 = C \) (where \( C \) is an arbitrary constant): \[ e^x + e^{-y} = C \] This represents the general solution, so Reason (R) is true.

Step 3: Evaluate Assertion (A).
A particular solution is obtained by substituting a specific value for the constant \( C \). If we select \( C = -2 \), the equation becomes: \[ e^x + e^{-y} = -2 \] This matches the assertion statement perfectly. Thus, Assertion (A) is true. Conclusion:
Both statements are true, and the general solution equation directly explains how the particular solution is constructed. Thus, option (A) is the correct choice.