Question

Consider the differential equation \[ \frac{dy}{dx} + y \ln(y) = 0 \] If \( y(0) = e \), then \( y(1) \) is _________

• A. ( e^e \)
• B. ( e^{-e} \)
• C. ( e^{(1/e)} \)
• D. ( e^{(-1/e)} \)

Hint:

For separable differential equations, separate the variables and integrate both sides to find the solution. Use initial conditions to determine constants.

Answer 1

The Correct Option is B

Step 1: Solving the differential equation.
The differential equation is separable, so we can rewrite it as: \[ \frac{dy}{y \ln(y)} = -dx \] Integrating both sides, we get: \[ \int \frac{1}{y \ln(y)} dy = \int -dx \] The integral on the left-hand side is a standard form, yielding: \[ \ln(\ln(y)) = -x + C \] Using the initial condition \( y(0) = e \), we can solve for \( C \): \[ \ln(\ln(e)) = 0 + C \Rightarrow C = 0 \] Thus, the solution is: \[ \ln(\ln(y)) = -x \] Exponentiating both sides: \[ \ln(y) = e^{-x} \] Taking the exponential of both sides: \[ y = e^{e^{-x}} \] Step 2: Finding \( y(1) \).
Substituting \( x = 1 \) into the solution: \[ y(1) = e^{e^{-1}} = e^{-e} \] Step 3: Conclusion.
Thus, the correct answer is (B) \( e^{-e} \).