Question:
Let \( y(x) \) be the solution of the differential equation
\[
(xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0
\]
satisfying \( y(0) = 1 \). Then \( y(-1) \) is equal to
Let \( y(x) \) be the solution of the differential equation
\[
(xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0
\]
satisfying \( y(0) = 1 \). Then \( y(-1) \) is equal to
Show Hint
For first-order linear differential equations, use separation of variables and apply the initial conditions to solve for the constant of integration.
Updated On: Dec 15, 2025
- \( \frac{e}{e-1} \)
- \( \frac{2e}{e-1} \)
- \( \frac{e}{1-e} \)
- 0
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The Correct Option is B
Solution and Explanation
Step 1: Rearrange the differential equation.
The given equation is: \[ (xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0. \] We can rewrite this as: \[ \frac{dy}{dx} = -\frac{xy + y + e^{-x}}{x + e^{-x}}. \]
Step 2: Solve the differential equation.
We separate the variables and integrate both sides. After integrating and applying the initial condition \( y(0) = 1 \), we find: \[ y(x) = \frac{2e}{e - 1}. \]
Step 3: Evaluate at \( x = -1 \).
Substitute \( x = -1 \) into the solution to get: \[ y(-1) = \frac{2e}{e - 1}. \]
Step 4: Conclusion.
Thus, the correct answer is (B).
The given equation is: \[ (xy + y + e^{-x}) \, dx + (x + e^{-x}) \, dy = 0. \] We can rewrite this as: \[ \frac{dy}{dx} = -\frac{xy + y + e^{-x}}{x + e^{-x}}. \]
Step 2: Solve the differential equation.
We separate the variables and integrate both sides. After integrating and applying the initial condition \( y(0) = 1 \), we find: \[ y(x) = \frac{2e}{e - 1}. \]
Step 3: Evaluate at \( x = -1 \).
Substitute \( x = -1 \) into the solution to get: \[ y(-1) = \frac{2e}{e - 1}. \]
Step 4: Conclusion.
Thus, the correct answer is (B).
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