Last month we talked about how to find numerically the solution of a differential equation. This method is called the Euler method. This method is particular useful in case where a differential equation is impossible to solve. The direction field method is a graphical approach to solve a differential equation. where there is no method to find an explicit solution for this equation.
Approach
Let's consider the differential equation y' = x + y with the initial value solution y(0) =1. We can draw small tangents at different points of the coordinate plane. Let's take the point (1,0) which is a solution to the differential equation. In order to draw a tangent line at this point we need to find the slope of the tangent line at that point. The slope is the derivative of the function at the point.
The slope at the point (1,0) is y' = 1 + 0 = 1. Let's draw the tangent line at this point.
. We can continue to draw tangent lines at different points of the coordinate plane. It's tedious to draw a big amount of tangent lines at many points of the coordinate plane. A computer program comes to the rescue to do this in a very short amount of time as shown by the figure below.
In this graph we can see different tangent lines that have positive, negative or null slope. The tangent lines oriented upward have a positive slope. Those oriented downward have a negative slope. Those that are parallel to the x-axis has a slope equal to zero. Some tangents are steeper than the other, For example the slope at the point (1.3) is steeper than the slope at the point (0,1). The steeper slopes tend to be more vertical compared to the others that are less steep.
This set of tangent lines is called direction field because it gives the direction where the solution curves are heading. The figure below shows a solution curve of the differential equation that passes through the point (0,1)
By following the direction of the tangent lines we can draw as many solution curves as possible.
