1. Let's start by understanding what a differential equation is. A differential equation is an equation that involves an unknown function and its derivatives. It describes how a quantity changes with respect to another, often time or space. 2. For example, a simple differential equation is $$\frac{dy}{dx} = y$$. This means the rate of change of $y$ with respect to $x$ is equal to $y$ itself. 3. The goal in solving a differential equation is to find the function $y(x)$ that satisfies the equation. 4. Differential equations can be ordinary (ODEs), involving derivatives with respect to one variable, or partial (PDEs), involving derivatives with respect to multiple variables. 5. A common method to solve simple ODEs like $$\frac{dy}{dx} = y$$ is separation of variables. We rewrite it as $$\frac{1}{y} dy = dx$$ and then integrate both sides. 6. Integrating, we get $$\int \frac{1}{y} dy = \int dx$$ which gives $$\ln|y| = x + C$$ where $C$ is the constant of integration. 7. Exponentiating both sides, we find $$y = Ce^{x}$$, which is the general solution to the differential equation. 8. In summary, differential equations describe relationships involving rates of change, and solving them means finding functions that satisfy those relationships. This is a basic introduction to what differential equations are and how you might start solving a simple one.