Question

Given the following conditions, determine the value of the 15-minute Peak Hourly Factor (PHF): Given:
- Vehicles in the peak hour come in 10-minute intervals
- Find the value of the 15-minute PHF

Hint:

When solving first-order differential equations, separating the variables and integrating both sides is a common method, especially when the equation can be rewritten into a product of functions of \( x \) and \( y \).

Answer 1

This is a first-order differential equation. To solve it, we will use the method of separation of variables. First, rewrite the equation as: \[ \frac{dy}{dx} = e^{x} \cdot e^{-y} \] Now, separate the variables: \[ e^{y} \, dy = e^{x} \, dx \] Integrating both sides: \[ \int e^{y} \, dy = \int e^{x} \, dx \] This gives: \[ e^{y} = e^{x} + C \] Where \( C \) is the constant of integration. Thus, the solution to the differential equation is: \[ e^{y} = e^{x} + C \]