Question

Given that dy/dx = ye^x such that x = 0, y = e. The value of y(y > 0) when x = 1 will be

• A. e
• B. 1/e
• C. e^e
• D. e²

Answer 1

The Correct Option is C

To solve the given differential equation and find the value of \( y \) when \( x = 1 \), we proceed as follows:

1. We are given the differential equation: \(\frac{dy}{dx} = y e^x\).
2. This is a separable differential equation. We can rewrite it as: \(\frac{dy}{y} = e^x dx\).
3. Integrate both sides: \(\int \frac{1}{y} \, dy = \int e^x \, dx\).
4. The left side integrates to: \(\ln |y|\), and the right side integrates to: \(e^x + C\) (where \( C \) is the integration constant).
5. The general solution of the differential equation is: \(\ln |y| = e^x + C\).
6. Exponentiating both sides to solve for \( y \), we get: \( y = e^{e^x + C} = e^C \cdot e^{e^x} \), let \( e^C = K \) for simplification.
7. The equation becomes: \( y = K \cdot e^{e^x} \).
8. Use the initial condition \( x = 0, y = e \) to find \( K \):
   • Substitute into the equation: \( e = K \cdot e^{e^0} = K \cdot e \).
   • Solving for \( K \) gives: \( K = 1 \).
9. The particular solution is: \( y = e^{e^x} \).
10. We need to find \( y \) when \( x = 1 \): \( y = e^{e^1} = e^e \).
11. Thus, the value of \( y \) when \( x = 1 \) is: \( e^e \).

The correct answer is e^e.