Phase line

Phase line

Geometrical and Numerical Methods for First-Order Equations

Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017

Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski

Note 2: The stability of y* cannot be determined by this theorem in the case when f′(y*) = 0. In this situation, the stability of y* is determined by the higher-order terms of the Taylor expansion. We would need to use other means, e.g., the phase line, in order to conclude anything about the stability of the equilibrium.

Qualitative aspects of differential equations in an inquiry-oriented course

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2023

Samer S. Habre

A central graphical approach emphasized in the IODE curriculum is the concept of slope fields. At any given point of the plane, the differential equation gives a value for the slope of the tangent to the solution curve passing through that point. A sketch of mini-tangents at various points of the plane constitutes the slope field (see a sample slope field in Figure 1). Starting at any given point (initial condition) and following the direction of the mini-tangents give a visual description of the solution through that point, a picture that portrays the increase and decrease of the solution, its concavity, its long term behaviour as . In the special case of an autonomous ODE , the phase line which is a one-dimensional projection of graphs of solution functions captures the entire qualitative behaviour of these solutions (see Figure 2).

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Geometrical and Numerical Methods for First-Order Equations

Qualitative aspects of differential equations in an inquiry-oriented course