Differential equations can be categorized in different ways based on their form and structure. Let's go over separable, linear, and Bernoulli differential equations with definitions and examples:

1. Separable Differential Equations

A separable differential equation can be written in the form:

$\frac{dy}{dx} = g(x)h(y)$

In this form, the variables $x$ and $y$ can be separated on opposite sides of the equation. You can solve by integrating both sides.

Example: $\frac{dy}{dx} = 3xy$ Here, the equation is separable, and we can rewrite it as: $\frac{1}{y} dy = 3x dx$ Then, integrate both sides: $\int \frac{1}{y} dy = \int 3x dx$ This leads to the solution: $\ln|y| = \frac{3x^2}{2} + C$ which simplifies to: $y = \pm e^{\frac{3x^2}{2} + C}$

2. Linear Differential Equations

A linear differential equation is one in which the dependent variable $y$ and its derivatives appear to the first power, with no products of $y$ and its derivatives. The standard form is:

$\frac{dy}{dx} + P(x)y = Q(x)$

Where $P(x)$ and $Q(x)$ are functions of $x$.

Example: $\frac{dy}{dx} + 2y = e^x$ Here, it's a linear first-order differential equation. To solve this, you would find an integrating factor, $\mu(x) = e^{\int P(x) dx}$, which simplifies the equation into an exact form.

3. Bernoulli Differential Equations

A Bernoulli differential equation has the form:

$\frac{dy}{dx} + P(x)y = Q(x)y^n$

Where $n$ is any real number. When $n = 1$, it becomes a linear differential equation. For other values of $n$, you can make a substitution to convert it to a linear equation.

Example: $\frac{dy}{dx} + y = y^2$ This is a Bernoulli equation where $n = 2$. You can solve it by making the substitution $v = y^{1-n} = y^{-1}$. After the substitution, it becomes linear: $-\frac{1}{y^2} \frac{dy}{dx} + \frac{1}{y} = 1$ Solving this gives you a solution for $y$.

---

Summary of the Key Types:

1. Separable Equation: Variables can be separated for integration.
2. Linear Equation: Linear in $y$ and its derivatives, with a standard form.
3. Bernoulli Equation: A nonlinear form that can be converted to a linear one using a substitution.

---

Would you like any further details or a deeper dive into how to solve these types?

Questions to Explore:

1. How do you solve a separable equation step by step?
2. What is an integrating factor in the linear equation method?
3. How do you handle special cases when solving Bernoulli equations?
4. How can you check whether a differential equation is separable or linear?
5. What are the practical applications of these types of differential equations?

Tip: Always check if a differential equation can be simplified or transformed into a standard form before trying to solve it.