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

• If you are given a set of initial conditions then there will be one solution trajectory

• To sketch a solution trajectory

  • Find the eigenvalues and eigenvectors of the matrix of coefficients (if needed)

    • They might be given in the question

  • Mark the starting point on the diagram

    • This will be given in the question

    • Or you will be given the solution and need to substitute $t=0$

  • Determine the initial direction of the trajectory

    • Find the values of $\frac{dx}{dt}$ when $t=0$ to determine if it is moving right or left

    • Find the values of $\frac{dy}{dt}$ when $t=0$ to determine if it is moving up or down

  • Sketch the trajectory using the general shape based on the eigenvalues

    • You should sketch lines for the eigenvectors if the eigenvalues are real