In this comprehensive exploration, we focus on separable differential equations. These equations, pivotal in modelling various real-world phenomena, require adept use of integration techniques for their solutions.

Understanding Separable Differential Equations

Separable differential equations can be written as $\frac{dy}{dx} = g(x)h(y)$, where $g(x)$ and $h(y)$ are functions solely of $x$ and $y$ respectively. The solution process involves separating the variables $x$ and $y$, and then integrating both sides of the equation.

Key Integration Techniques

To solve these equations, students must be familiar with three key integration techniques: integration by substitution, integration by parts, and the method of partial fractions. Each technique is suited to different types of differential equations and is crucial for finding the general solution.

Examples Illustrating These Methods

Example 1: Basic Separation of Variables

Problem: Solve $\frac{dy}{dx} = x^2y$.

Solution:

1. Separate Variables: Rearrange to $\frac{1}{y}dy = x^2dx$.

2. Integrate Both Sides:

• Integrate $x^2$ with respect to $x$, giving $\frac{x^3}{3}$.
• Integrate $\frac{1}{y}$ with respect to $y$, giving $\ln|y|$.

3. Combine Results: The solution is $\ln|y| = \frac{x^3}{3} + C$, where $C$ is the constant of integration.

Example 2: Integration by Parts

Problem: Solve $\frac{dy}{dx} = \frac{e^x}{y}$.

Solution:

1. Rearrange Equation: Convert to $ydy = e^xdx$.

2. Integrate Both Sides:

• Integrate $e^x$ with respect to $x$, yielding $e^x$.
• Integrate $ydy$, resulting in $\frac{y^2}{2}$.

3. Combine Results: The general solution is $\frac{y^2}{2} = e^x + C$.

Example 3: Using Partial Fractions

Problem: Solve $\frac{dy}{dx} = \frac{4x}{(x^2+1)y}$.

Solution:

1. Rearrange Equation: Modify to $ydy = \frac{4x}{x^2+1}dx$.

2. Integrate Using Partial Fractions:

• Integrate $\frac{4x}{x^2+1}$ with respect to $x$, which results in $2\ln|x^2+1|$.
• Integrate $ydy$, yielding $\frac{y^2}{2}$.

3. Combine Results: The solution is $\frac{y^2}{2} = 2\ln|x^2+1| + C$.