Math Insight

Introduction to autonomous differential equations

Video introduction

Overview of autonomous differential equation

An autonomous differential equation is an equation of the form \begin{align*} \diff{y}{t} = f(y). \end{align*} Let's think of $t$ as indicating time. This equation says that the rate of change $dy/dt$ of the function $y(t)$ is given by a some rule. The rule says that if the current value is $y$, then the rate of change is $f(y)$.

The equation is called a differential equation, because it is an equation involving the derivative $dy/dt$. The differential equation is called autonomous because the rule doesn't care what time $t$ it is. It only cares about the current value of the variable $y$.

Given an autonomous differential equation, we'll often want to solve the equation, which means find a function a $y(t)$ whose derivative $dy/dt$ is equal to $f(y)$.

A linear differential equation

An example of an autonomous differential equation, let's let $f(y)$ be the linear function $f(y)=2y$, so the equation is \begin{align*} \diff{y}{t} = 2y. \end{align*} which we could also write as \begin{align*} \diff{y}{t}(t) = 2y(t), \end{align*} but, usually, we won't write out the explicit dependence on $t$.

We've ended up with a linear differential equation, one of the simplest types.

Solution methods

How can we solve this equation, i.e., find a function $y(t)$ that, if we differentiate, we get the function $y(t)$ back, only multiplied by two? We have three main methods for solving autonomous differential equations.

1. Numerical methods. We can approximate the continuous change of the differential equation with discrete jumps in time, By doing this, we get a formula for evolving from one time step to the next (like a a discrete dynamical system). We can then use a computer to calculate this approximation to the solution.
2. Graphical methods. We can use a plot of the graph of $f(y)$ to determine the behavior of $y(t)$.
3. Analytic methods. We can use mathematics to find a function $y(t)$ that satisfies the differential equation.

The Guess and Check method

Here, let's try an analytic method. We'll use an important analytic method, which is called guess and check. Guess and check is a perfectly valid method because, if you can find any function that satisfies the equation, you've found a solution. It doesn't matter what method you use to find the function; as long as you can show it is a solution, you've accomplished the task.

The way to guess is to use your knowledge and intuition above derivatives to find a function whose derivative behaves in the required way.

Here we want to find a function where, if we take the derivative, we get the function back, only multiplied by 2. Let's forget about the factor of two for a moment. Do you know any function that is it's own derivative?

The exponential function fits the bill. Remember that \begin{align*} \diff{}{t} e^t = e^t. \end{align*} so if $y(t)=e^t$, then $\diff{y}{t} = e^t = y(t)$.

That's pretty close to the right behavior. We just want the derivative to bring down a factor of two. Remember the chain rule and how the chain rule says that the derivative of $f(g(t))$ gives the derivative of $f$ multiplied by the derivative of $g$. If we make $g(t)=2t$, then its derivative is 2, so we multiply by 2. In fact, if \begin{align*} y(t) = e^{2t} \end{align*} then \begin{align*} \diff{y}{t} = 2e^{2t} = 2y(t). \end{align*} That's exactly what we are looking for. The function $y(t)=e^{2t}$ solves the differential equation \begin{align*} \diff{y}{t} = 2y. \end{align*} Our guess and check method found the solution.

The general solution

We found a solution OK, but is $y(t)=e^{2t}$ the only solution? Is there any other function whose derivative is twice the function?

What if we multiplied $y(t)$ by 3? We know that if we multiply a function by a constant number, that number just comes out of the derivative. Let's try the function \begin{align*} y(t) = 3e^{2t}. \end{align*} With that $y(t)$, \begin{align*} \diff{y}{t} = 6 e^{2t} = 2(3 e^{2t}) = 2y(t). \end{align*} So the function $y(t)=3e^{2t}$ works. In fact so does $5e^{2t}$, $2.3 e^{2t}$ or $Ce^{2t}$ for any number $C$. In general any function of the form \begin{align*} y(t) = Ce^{2t} \end{align*} will satisfy the differential equation $\diff{y}{t} = 2y$. The constant number $C$ just comes out of the derivative. It turns out that all solutions are of this form. We call $y(t)$, the general solution to the differential equation.