### Introduction to autonomous differential equations

#### Video introduction

*Introduction to autonomous differential equations.*

#### 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.

**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.**Graphical methods.**We can use a plot of the graph of $f(y)$ to determine the behavior of $y(t)$.**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.

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