How to Solve Variable Separable Differential Equations

Variable separable differential equations are a fundamental topic in calculus, often encountered in various scientific and engineering applications. One such equation is:

[frac{dy}{dx} ae^{y} - b]

This can be solved by separating the variables and integrating both sides. Here, we will walk through the steps and provide an in-depth analysis of the solution behavior.

Step-by-Step Solution

First, we separate the variables:

[frac{dy}{ae^{y} - b} dx]

Next, we integrate both sides:

[int frac{dy}{ae^{y} - b} int dx]

The left-hand side integral can be solved by using a substitution. Let:

[ae^{y} - b t]

Then:

[ae^{y}dy dt Rightarrow dy frac{dt}{ae^{y}}]

Substitute back:

[int frac{1}{ae^{y} - b} cdot ae^{y} frac{dt}{a} int dx]

Let:

[ae^{y} u Rightarrow e^{y} frac{u}{a}]

Then:

[int frac{1}{u - b} frac{du}{a} int dx]

Integrating both sides:

[frac{1}{a} int frac{du}{u - b} x C]

Yielding:

[frac{1}{a} ln left|frac{u}{u - b}right| x C]

Substituting back:

[frac{1}{a} ln left|frac{ae^{y} - b}{ae^{y} - b - b}right| x C]

Simplifying:

[frac{1}{a} ln left|frac{ae^{y}}{ae^{y} - b}right| x C]

Which simplifies to:

[frac{1}{b} ln left|frac{ae^{y}}{ae^{y} - b}right| x C]

Therefore, the solution to the differential equation is:

[ln left|frac{ae^{y}}{ae^{y} - b}right| bx C]

Behavior and Analysis

To better understand the behavior of the solution, let's consider some special cases.

Case 1: When (b 0)

For (b 0), the equation simplifies to:

[frac{dy}{dx} ae^{y}]

Integrating both sides, we get:

[frac{dy}{ae^{y}} dx]

Which yields:

[frac{1}{a} ln left|e^{y}right| x C]

Or:

[ln left|e^{y}right| ax C]

Thus:

[y ln (e^{ax C}) ax C]

Where (C) is a constant.

Case 2: Analyzing Super-Exponential Growth

The general equation [frac{dy}{dx} ae^{y} - b] can exhibit super-exponential growth or decrease depending on the values of (a) and (b). For instance, if (a > 0), the function can grow faster than any polynomial or exponential function.

To investigate this, consider successive differentiation and substitution for (f(x)). By differentiating and substituting, we can determine:

[f^{(n)}(x) (n-1)! a^{n} e^{n f(x)}]

For an algebraic Maclaurin expansion about the origin:

[f(x) f(0) sum_{n1}^{infty} frac{a e^{f(0)} x^n}{n}]

The series converges when (|a e^{f(0)} x|

For a series sum, we have:

[f(x) f(0) - ln(1 - a e^{f(0)})x]

This solution satisfies the differential equation for the range of (x) given earlier.

Conclusion

Variable separable differential equations are not only useful in theoretical studies but also in practical applications. The solution methods and the analysis of their behavior provide valuable insights into complex systems.

Keywords: variable separable differential equation, exponential growth, numerical integration, solution scaling.