11.8 (Systems of Differential Equations), Taylor Approximations for DEs Flashcards
What is a system of differential equations?
A system of differential equations is a set of two or more equations involving derivatives of multiple variables that depend on each other.
When is a system called coupled?
A system is called coupled when the equations involve interactions between multiple variables, meaning the derivative of one variable depends on the other variable(s).
what are the methods to solve a system of differential equations?
Systems can often be solved using substitution, graphical methods, or matrix techniques like eigenvalues and eigenvectors if the system is linear.
What defines a linear system of differential equations?
A linear system has equations that are linear in terms of the variables and their derivatives
What does it mean if a system has a steady state?
A steady state is a point where the derivatives (changes) are zero, meaning the system reaches an equilibrium and the variables stop changing over time.
What is the purpose of Taylor approximation for differential equations?
Taylor approximation uses a Taylor series expansion to approximate solutions of differential equations around a point, often near an initial condition.
What information is needed to construct a Taylor approximation for a differential equation?
The initial value y(a), and values of successive derivatives 𝑦′(𝑎) etc., at the point x=a.
