1. The problem is to understand what differential equations are and how to solve a simple example. 2. A differential equation is an equation that relates a function with its derivatives. It shows how a quantity changes with respect to another, often time or space. 3. The general form of a first-order differential equation is $$\frac{dy}{dx} = f(x,y)$$ where $y$ is the function of $x$. 4. Let's solve a simple differential equation: $$\frac{dy}{dx} = 3y$$. 5. This is a separable differential equation. We can rewrite it as $$\frac{1}{y} dy = 3 dx$$. 6. Integrate both sides: $$\int \frac{1}{y} dy = \int 3 dx$$. 7. The integrals give: $$\ln|y| = 3x + C$$ where $C$ is the constant of integration. 8. Exponentiate both sides to solve for $y$: $$y = e^{3x + C} = e^C e^{3x}$$. 9. Let $A = e^C$, a constant, so the solution is $$y = A e^{3x}$$. 10. This means the function $y$ grows exponentially with rate 3. This is a basic example of solving a first-order differential equation by separation of variables.