1. **Stating the problem:** Solve the differential equation given (though the exact equation is not specified, we assume a common form such as $\frac{dy}{dx} = f(x,y)$). 2. **General approach:** To solve a differential equation, we often use methods like separation of variables, integrating factors, or characteristic equations depending on the type. 3. **Example:** Suppose the differential equation is $\frac{dy}{dx} = ky$ where $k$ is a constant. 4. **Formula used:** The solution to this first-order linear differential equation is given by $$y = Ce^{kx}$$ where $C$ is the integration constant. 5. **Explanation:** This comes from separating variables: $$\frac{dy}{y} = k \, dx$$ Integrate both sides: $$\int \frac{1}{y} dy = \int k \, dx$$ $$\ln|y| = kx + C_1$$ Exponentiate both sides: $$|y| = e^{kx + C_1} = e^{C_1} e^{kx}$$ Let $C = e^{C_1}$, so $$y = Ce^{kx}$$ 6. **Final answer:** The general solution is $$y = Ce^{kx}$$ where $C$ is an arbitrary constant determined by initial conditions.