Post 4: Applications of Separable Differential Equations

Real-World Applications

Separable differential equations have a wide range of applications in various fields, including physics, biology, and economics. These equations are used to model and predict outcomes in different scenarios by expressing the rate of change of a dependent variable in terms of its independent variable. Let's explore some examples of how separable differential equations are applied in real-world contexts.

Physics Example: Radioactive Decay

One common application of separable differential equations is in modeling radioactive decay. The rate of decay of a radioactive substance is proportional to the amount of the substance present at any given time. This relationship can be modeled using a separable differential equation.

Let's denote the amount of the radioactive substance at time t as N(t). The rate of decay, dN/dt, is proportional to N(t) and can be expressed as:

dN/dt = -kN(t)

Where k is the decay constant. This equation is separable, as we can separate the variables by rearranging it as:

1/N(t) * dN = -k dt

Now, we can integrate both sides of the equation:

∫1/N(t) * dN = ∫-k dt

Integrating, we obtain:

ln|N(t)| = -kt + C

Where C is the constant of integration. By solving for N(t), we can determine the amount of the radioactive substance at any given time.

Biology Example: Population Growth

Another application of separable differential equations is in modeling population growth. Let's consider a simple population model where the rate of change of a population, dP/dt, is proportional to the size of the population at any given time, P(t). This relationship can be described using a separable differential equation.

The equation can be written as:

dP/dt = kP(t)

Where k is the growth rate constant. By rearranging the equation, we have:

1/P(t) * dP = k dt

Integrating both sides:

∫1/P(t) * dP = ∫k dt

This simplifies to:

ln|P(t)| = kt + C

Solving for P(t), we can determine the population size at any given time.

Economics Example: Compound Interest

Compound interest is another area where separable differential equations find application. Consider an investment that earns compound interest at a fixed rate. The amount of money, A(t), in the account at time t can be modeled using a separable differential equation.

The equation can be written as:

dA/dt = rA(t)

where r is the interest rate. By separating variables, the equation becomes:

1/A(t) * dA = r dt

After integrating both sides:

ln|A(t)| = rt + C

Solving for A(t), we can determine the value of the investment at any given time.

Conclusion

Separable differential equations are powerful tools for modeling and predicting various real-world phenomena. Whether it's radioactive decay, population growth, or compound interest, separable differential equations allow us to express the relationship between variables and understand how they change over time. By solving these equations, we can make informed decisions and predictions in various fields of study.