1. The problem is to understand what a differential equation is and how to solve a simple example. 2. A differential equation is an equation involving derivatives of a function. It relates a function with its rates of change. 3. For example, consider the simple differential equation $$\frac{dy}{dx} = ky$$ where $k$ is a constant. 4. To solve it, separate variables: $$\frac{1}{y} dy = k dx$$. 5. Integrate both sides: $$\int \frac{1}{y} dy = \int k dx$$, which gives $$\ln |y| = kx + C$$ where $C$ is the integration constant. 6. Exponentiate both sides to solve for $y$: $$|y| = e^{kx + C} = e^{C} e^{kx}$$. 7. Let $A = e^{C}$ (a positive constant), so $$y = A e^{kx}$$. 8. This is the general solution to the differential equation $$\frac{dy}{dx} = ky$$. 9. If an initial condition is given, such as $y(x_0) = y_0$, then solve for $A$: $$y_0 = A e^{k x_0} \Rightarrow A = y_0 e^{-k x_0}$$. 10. Thus, the specific solution becomes $$y = y_0 e^{k(x - x_0)}$$. This method applies to first order linear differential equations with separable variables.