Solving the Differential Equation x1 dy/dx 2e-y1: A Comprehensive Guide

The differential equation x1 dy/dx 2e-y1 is a first-order, separable differential equation. Such equations are fundamental in various fields of science and engineering, particularly where modeling exponential growth or decay is necessary.

Motivation and Context

Understanding how to solve such differential equations is essential for studying phenomena that change over time. This equation models scenarios such as population growth, radioactive decay, or chemical reactions where the rate of change depends exponentially on the current state of the system.

The Equation in Detail

Let's break down and solve the differential equation: x1 dy/dx 2e-y1. To make it easier to understand, let's express it in a more formal mathematical format:

[frac{x_1dy}{dx} 2e^{-y}1]

This equation is of the separable variables type. Let's separate the variables on each side of the equation:

[frac{dy}{2e^{-y}1} frac{dx}{x_1}]

Integration Step

Now, we will integrate both sides of the equation:

[int frac{dy}{2e^{-y}1} int frac{dx}{x_1}]

Let's focus on the left-hand side:

[int frac{dy}{2e^{-y}1} int frac{e^y,dy}{2e^y}]

Notice that (2e^y) in the denominator and numerator will cancel out, simplifying the integral:

[int frac{e^y,dy}{2e^y} int frac{dy}{2} frac{y}{2} C_1]

Now, for the right-hand side:

[int frac{dx}{x_1} frac{x}{x_1} C_2]

Equating these integrals, we get:

[frac{y}{2} C_1 frac{x}{x_1} C_2]

Solving for y

Let's set (C_1 - C_2 C) for simplicity:

[frac{y}{2} frac{x}{x_1} C]

Multiplying both sides by 2:

[y frac{2x}{x_1} 2C]

Let's denote the constant (2C) as (K):

[y frac{2x}{x_1} K]

Exponential Form

For certain applications, especially when dealing with exponential growth or decay, the solution can be more conveniently expressed in the form of an exponential function:

[e^y Ke^{frac{2x}{x_1}}]

By taking the natural logarithm of both sides:

[y ln(Ke^{frac{2x}{x_1}}) ln(K) frac{2x}{x_1}]

Here, (ln(K)) is a constant, and we can denote it as (K'):

[y K' frac{2x}{x_1}]

Conclusion

In summary, the solution to the differential equation x1 dy/dx 2e-y1 is:

[y K' frac{2x}{x_1}]

This solution is a general form that captures the behavior of the system under study. It can represent various physical, biological, or chemical processes depending on the context.

Further Exploration

For those interested in more advanced topics, you might want to explore:

Non-separable differential equations Systems of differential equations Applications of differential equations in engineering and science

Keywords

Differential Equation, Separable Variables, Integration, Exponential Functions