19.2: Differential Equations

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A differential equation (DE) is an equation that describes the derivatives of an unknown function. “Solving a DE” means finding a function whose derivatives satisfy the equation.

For example, suppose we would like to predict the population of yeast growing in a nutrient solution. Assume that we know the initial population is 5 billion yeast cells. When yeast grow in particularly yeast-friendly conditions, the rate of growth at any point in time is proportional to the current population. If we define \(y(t)\) to be the population at a time \(t\), we can write the following equation for the rate of growth: \[\frac{dy}{dt}(t) = a y(t)\notag\] where \(\frac{dy}{dt}(t)\) is the derivative of \(y(t)\) and \(a\) is a constant that characterizes how quickly the population grows. This equation is differential because it relates a function to one of its derivatives.

It is an ordinary differential equation (ODE) because all the derivatives involved are taken with respect to the same variable. If it related derivatives with respect to different variables (partial derivatives), it would be a partial differential equation (PDE).

This equation is first order because it involves only first derivatives. If it involved second derivatives, it would be second order, and so on.

Lastly, it’s linear because each term involves \(t\), \(y\), or \(dy/dt\) raised to the first power; if any of the terms involved products or powers of \(t\), \(y\), or \(dy/dt\) it would be nonlinear.

Now suppose we want to predict the yeast population in the future. We can do that using Euler’s method.