Main Concept

A differential equation (DE) is a mathematical equation which relates an unknown function and one or more of its derivatives. DEs are an important tool in many areas of pure and applied mathematics, as well as other subjects such as physics, engineering, economics, and biology. The order of a DE is the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation.

The focus of this app is on first-order differential equations, which only include the first derivative and the function itself.

These DEs often have the form $y \' \= f\left(x\,y\right)$ where x is the independent variable, y is the dependent variable, and so  $y \'$, also written$\frac{\ⅆ}{\ⅆ x} y\left(x\right)$, is the derivative of the function $y\left(x\right)$ with respect to x.

Although there are many techniques for solving first-order DEs, it can be very difficult or even impossible to solve them explicitly in some cases, and so there is need for a way to visualize and numerically approximate the solution curves. This is where direction fields become useful.

A direction field (or slope field) is a graphical representation of the solutions of a first-order differential equation achieved without solving the DE explicitly. At each point (x,y) in the plane, you plot the direction vector $\left[\begin{array}{c}1\\ f\left(x\,y\right)\end{array}\right]$, which is tangent to (has the same slope as) the solution curve through that point. These vectors are often normalized to be of consistent length to emphasize their direction rather than magnitude. You can then approximate the solution curves by following the direction vectors.

Often, you are also given an initial condition,  $y\left(x_0\right) \= y_0$. This allows you to illustrate a specific curve in the family of solutions by ensuring that the curve you sketch passes through the point $\left(x_0 \, y_0\right)$ while still following in the directions of the vectors.

Initial Condition: $y\($$\) \=$