Show me how to solve differentiable equations, and how they relate to optics

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Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental tools for modeling systems that change over time or space. For example, the equation dy/dx equals y describes exponential growth, where the rate of change is proportional to the current value. Let's solve the differential equation dy/dx equals ky step by step. First, we separate variables by dividing both sides by y and multiplying by dx. Next, we integrate both sides to get the natural log of y equals kx plus a constant. Finally, we solve for y by exponentiating both sides, giving us the general solution y equals C times e to the kx. Maxwell's equations are four fundamental differential equations that govern all electromagnetic phenomena, including light. These equations describe how electric and magnetic fields interact and change over time. When combined, they lead to the wave equation, which shows that light is an electromagnetic wave traveling at the speed of light. This mathematical framework is essential for understanding all optical phenomena. When light waves encounter obstacles or apertures, they diffract according to the wave equation. The wave equation is a differential equation that describes how electromagnetic fields propagate through space. By solving this equation with specific boundary conditions defined by apertures or obstacles, we can predict the resulting diffraction patterns. This demonstrates how differential equations directly govern optical phenomena like the bending and interference of light. Differential equations are essential for designing and analyzing optical systems. They enable engineers to optimize lens designs, model fiber optic propagation, analyze laser cavities, and calculate holographic patterns. From Snell's law derived from Fermat's principle to the wave equation governing light propagation, differential equations provide the mathematical foundation for all modern optical and photonic technologies.