Slope Fields (DP IB Applications & Interpretation (AI)): Revision Note

Slope fields

What are slope fields?

• Slope fields are used to visualise the solutions to a different equation

  • They can be drawn without knowing the actual solutions

• Consider the differentiation equation

• You can find the value of at various points

  • Just substitute values for and into

• Each value is the gradient of a solution curve that goes through that point

• A slope field for a differential equation is a diagram with short tangent lines drawn at a number of points

  • 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?

• The tangent lines in a slope field diagram 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

How can I use slope fields to sketch a solution to a differential equation?

• You will be given a point that lies on the solution curve

  • Plot this point

• Use the tangent line at that point to determine the direction of the solution curve

  • You can go to the right and to the left

• Keep using the tangent lines to sketch the shape of the solution

  • You will only be expected to sketch the solution and not draw it accurately

• The shape of the solution curve depends on the given point

  • Different points lead to different shapes

What key features can be found using slope field diagrams?

• You can find the location of stationary points

  • These occur at horizontal tangent lines

  • These could be local minimums, local maximums or points of inflection

• You can find the exact points where stationary points lie by solving

• For example, consider if

  • when

  • Stationary points lie on the lines