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 $\frac{y^2}{2} = 2\ln|x^2+1| + C$.