5.6.3 Slope Fields

What are slope fields?

• We are considering here a differential equation involving two variables of the form 
• I.e., the derivative  is equal to some function of x and y
• In some cases it may be possible to solve the differential equation analytically, while in other cases this is not possible
• Whether or not the equation can be analytically solved, however, it is always possible to calculate the derivative  at any point (x, y) by putting the x and y values into 
• This means that we can calculate the gradient of the solution curve at any point that the solution might go through
• A slope field for a differential equation is a diagram with short tangent lines drawn at a number of points
• The gradient of the tangent line drawn at any given point will be equal to the value of  at that point
• Normally the tangent lines will be drawn for points that form a regularly-spaced grid of x and y values

How can I use slope fields to study the solutions of a differential equation?

• Looking at the tangent lines in a slope field diagram will give you a general sense for what the solution curves to the differential equation will look like
  • Remember that the solution to a given differential equation is actually a family of solutions
  • We need appropriate boundary conditions or initial conditions to determine which of that family of solutions is the precise solution in a particular situation
• You can think of the tangent lines in a slope diagram as ‘flow lines’
  • From a given point the solution curve through that point will ‘flow’ away from the point in the direction of the tangent line
• For a given point, you can use a slope field to sketch the general shape of the solution curve that goes through that point
  • The given point here serves as a boundary condition, letting you know which of all the possible solution curves is the one you want to sketch
  • The sketch should go through the given point, and follow the general ‘flow’ of the tangent lines through the rest of the slope field diagram
  • In general, the sketched solution curve should not attempt to connect together a number of different tangent lines in the diagram
    • There is no guarantee that the solution curve will go through any exact point in the ‘grid’ of points at which tangent lines have been drawn
    • The only tangent line that your solution curve should definitely go through is one at the given ‘boundary condition’ point
  • The sketched solution curve may go along some of the tangent lines, but it should not should not cut across any of them

• Look out for places where the tangent lines are horizontal
• At such points 
• Therefore such points may indicate local minimum or maximum points for a solution curve
• Be careful – not every point where  is a local minimum or maximum
• But every local minimum or maximum will be at a point where 
• Don’t forget that you can also solve the equation  directly to identify points where the gradient is zero
• For example if , then the gradient will be zero anywhere where 
• This is another way to identify possible local minimum and maximum points for the solution curves
• If such a point falls between the ‘grid points’ at which the tangent lines have been drawn, this may be the only way to identify such a point exactly