Differential Equations | Class 12 Mathematics Notes

Applications of Differential Equations

This section demonstrates how differential equations are applied to solve real-world problems. It discusses several examples from physics and biology where differential equations model natural phenomena. One classical example is Newton's law of cooling, which states that the rate of change of temperature of a body is proportional to the difference between its temperature and the ambient temperature. This leads to the differential equation dy/dt = -k(y - T), where y is the temperature of the body, T is the ambient temperature, and k is a positive constant. Solving this first-order linear differential equation provides the temperature as a function of time. Another example is the population growth model, where the rate of change of population is proportional to the current population, leading to the equation dy/dt = ky. The solution y = Ce^(kt) describes exponential growth or decay depending on the sign of k. The section also covers problems involving mixing solutions, radioactive decay, and motion under resistance, illustrating the versatility of differential equations. Step-by-step solutions show how to translate physical conditions into differential equations and solve them.

📊 Diagram: Diagrams include temperature vs. time graphs showing cooling curves, population growth curves, and schematic diagrams of tanks for mixing problems.

🧪 Activity: No specific activity, but examples serve as applied problem-solving exercises.

🔗 Connection: This section consolidates the understanding of differential equations by applying solution methods to real-world scenarios, reinforcing the importance of the concepts learned.