This section contains:

When you start learning how to integrate functions, you’ll probably be introduced to the notion of **Differential Equations** and **Slope Fields**.

# Differential Equations and Separation of Variables

A **differential equation** is basically any equation that has a derivative in it. For example, it typically contains \(x\), \(y\), and the **derivative of** \(y\) (\({y}’\) or \(\displaystyle \frac{{dy}}{{dx}}\)).

Remember the following from the **Antiderivatives and Indefinite Integration** section:

**Definitions of Antiderivative (Integral):**

A function \(F\) is an antiderivative of another function \(f\) when \({F}’\left( x \right)=f\left( x \right)\).

Note that the term **indefinite integral** is another word for an antiderivative, and is denoted by the integral sign \(\int{.}\)

When we have the **differential equation **(an equation that involve \(x\), \(y\) and the derivative of \(y\)) in the form \(\displaystyle \frac{{dy}}{{dx}}=f\left( x \right)\), we can write it as \(dy=f\left( x \right)dx\). When we integrate, we have \(\displaystyle y=\int{{f\left( x \right)dx}}=F\left( x \right)+\,\,C\), where \(C\) is the constant of integration.

But the most important thing to remember is that **integration is the opposite of differentiation**!

To solve a differential equation, we need to separate the variables so that one variable is on the left hand side of the equal sign (usually the “\(y\)”, including the “\(dy\)”), and the other variable is on the right hand side (usually the “\(x\)”, including the “\(dx\)”).

At this point, we can integrate on both sides, and put the \(“+\,C”\) on the side with the \(x\)’s (right away). We will simplify as needed, being careful to preserve the \(“+\,C”\) during the simplification. Notice that if we add, subtract, multiply, or divide with the \(“+\,C”\) when simplifying, we still end up with a constant \((+\,C\)).

Here are some examples:

Sometimes we have to solve the differential equation and then put in an initial value to get the \(“C”\). When doing this, it’s best to put in the initial condition immediately after differentiating. Here are some examples; sorry the math is so messy:

# Slope Fields

**Slope Fields** are a strange concept (they look funny!), but they really aren’t that difficult.

Slope fields are little lines on a coordinate system graph that represent the **slope **for that \((x,y)\) combination for a particular differential equation (remember that a differential equation represents a slope). For example, for the differential equation \(\displaystyle \frac{{dy}}{{dx}}=x+y\), for point \((0,0)\) on the slope field graph, the little line would be horizontal, since \(0+0=0\), and the slope of **0** is represented by a horizontal line.

Let’s show an example. If we have the differential equation \(\displaystyle \frac{{dx}}{{dy}}=\frac{x}{y},\) do you see how the slope will be **0** everywhere \(x\) is **0**, and it will be **1** everywhere \(x\) and \(y\) are the same? And it will be **2** when \(x\) is twice the value of \(y\), and so on? The slope is undefined (vertical lines or no lines) when \(y=0\).

So here’s what the slope field graph looks like. Again, remember that the little lines represent the slope, since a differential equation is a slope.

Here are more examples of slope fields. Note that if we solved the differential equation, we’d see the solution to that differential equation in the slope field pattern. For example, for the differential equation \(\displaystyle \frac{{dy}}{{dx}}=2\), the little lines in the slope field graph are \(\displaystyle y=2x\).

Here’s a few more that we’ll draw with solution curves, given the solution passes through the given initial points. To draw the solution curves, start with the initial point, and then follow the curve of the little lines as you best can:

**Learn these rules, and practice, practice, practice!**

On to **L’Hopital’s Rule** – you are ready!