Sketching Solution Trajectories (DP IB Applications & Interpretation (AI)): Revision Note

Sketching Solution Trajectories

How do I sketch a solution trajectory for a system of coupled differential equations?

• A phase portrait shows typical trajectories representing all the possible solutions to a system of coupled differential equations

• For a given set of initial conditions, however, the solution will only have one specific trajectory

• Sketching a particular solution trajectory will generally involve the following:

  • Make sure you know what the ‘typical’ solutions for the system look like

    • You don’t need to sketch a complete phase portrait unless asked, but you should know what the phase portrait for your system would look like

    • If the phase portrait includes ‘eigenvector lines’, however, it is worth including these in your sketch to serve as guidelines

  • Mark the starting point for your solution trajectory

    • The coordinates of the starting point will be the and values when

    • Usually these are given in the question as the initial conditions for the system

  • Determine the initial direction of the solution trajectory

    • To do this find the values of  and  when

    • This will tell you the directions in which and are changing initially

    • For example if  and  when , then the trajectory from the starting point will initially be ‘to the left and up’, parallel to the vector 

  • Use the above considerations to create your sketch

    • The trajectory should begin at the starting point (be sure to mark and label the starting point on your sketch!)

    • It should move away from the starting point in the correct initial direction

    • As it moves further away from the starting point, the trajectory should conform to the nature of a ‘typical solution’ for the system