IB Maths AI 5.9 Definitions

This page contains our IB Maths AI definitions for 5.9. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Advanced differential equations'.

All Topic 5 Sub-topics 5.1: Introduction To Derivatives 5.2: Applying Derivatives 5.3: Fundamentals Of Integration (HL)5.4: Advanced derivatives (HL)5.5: Further integration (HL)5.6: Applying integration (HL)5.7: Kinematics (HL)5.8: Introduction to differential equations (HL)5.9: Advanced differential equations

eigenvalues

Numbers associated with the matrix $A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}$ that determine the qualitative behaviour of solutions, such as whether paths move toward, away from, or around an equilibrium.

eigenvectors

Non-zero direction vectors corresponding to eigenvalues that give the principal directions of motion in a linear phase portrait.

equilibrium

A point where the system does not move because both derivatives are zero, so $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$ .

Euler

A numerical method that generates successive approximations to a solution by stepping forward with a fixed step size

h

using update rules such as

x_{n+1}=x_n+h y_n

phase

The angle that determines the horizontal shift of a sinusoidal function; in

a\cos\left(\omega t+\alpha\right)

it is

\alpha

saddle

An unstable equilibrium where one eigenvalue is positive and the other is negative, so some trajectories approach the equilibrium while others move away.

spiral

A phase-portrait pattern produced by complex eigenvalues, where trajectories curve around the equilibrium while moving inward or outward depending on the sign of the real part.

trajectories

Curves in a phase portrait that show how the state

(x,y)

changes over time for a particular solution.

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