In this unit we study systems of differential equations. A system of ODE’s means
a DE with one independent variable but more than one dependent variable, for
example:
\({\displaystyle x’ = x + y}\), \({\displaystyle y’ = x^2 - y - t}\)
is a 2x2 system of DE’s for the two functions \(x = x(t)\) and \(y = y(t)\).
As usual, we start with the linear case. Even for linear systems, though, it
turns out that efficient solution methods require some new techniques, namely
the machinery of matrix-vector algebra. A small investment in this background
material yields an excellent return, giving both the linear theory in the
general case and also the explicit computational methods for the solutions in
the constant-coefficient case.
We finish this unit by showing some of the qualitative theory of DE’s for
systems, linear and non-linear. Qualitative theory means finding out information
about the solutions directly from the DE without actually having to solve it. We
start with the linear case, and then show how we can use the results for linear
constant-coefficient systems to gain information about certain non-linear
systems using a technique called linearization.
- Captured On
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- Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare
Captured On [2020-01-21 Tue 13:07] Source Linear Systems | Unit IV First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 Can we solve a linear system of ODE’s with constant coefficients by eliminating variables? 1.1 Front Can we solve a linear system of ODE’s with constant coefficients by eliminating variables?
1.2 Back No, it’s a naive way to solve it. You need to use techniques of constant coefficient ODE methods....
Captured On [2020-02-01 Sat 12:35] Source Matrix Methods: Eigenvalues and Normal Modes | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the trace of a square matrix? 1.1 Front What is the trace of a square matrix?
For example, \({\displaystyle \operatorname{tr}\begin{pmatrix}a & b \\ c & d\end{pmatrix}}\)
1.2 Back It’s the sum of the elements on the main diagonal; it’s denoted \(\operatorname{tr}(A)\):...
Captured On [2020-02-13 Thu 17:57] Source Qualitative Behavior: Phase Portraits | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the phase plane for this system? 1.1 Front What is the phase plane for this system?
\({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \({\displaystyle A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}}\)
1.2 Back It’s the \(xy\text{-plane}\) itself, where you can draw the trajectory of a solution with an arrow to indicate the direction of increasing time....
Captured On [2020-02-20 Thu 21:51] Source Matrix Exponentials | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 For this linear system, how many solutions are there? 1.1 Front For this linear system, how many solutions are there?
\({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \(A\) is a \(n \cross n\) matrix
1.2 Back There are \(n\) linearly independent solutions for the system
2 How is the linear system when the coefficients are functions of the independent variable t?...
Captured On [2020-03-06 Fri 14:04] Source Nonlinear Systems | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an autonomous system? 1.1 Front What is an autonomous system?
General First Order Autonomous Systems
1.2 Back The word autonomous means self regulating. These systems are self regulating in the sense that their rate of change (e.g. derivatives) depends only on the state of the system (values of \(x\) and \(y\)) and not on the time \(t\)...
Captured On [2020-03-09 Mon 12:02] Source Linearization Near Critical Points | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 How can we get information about the trajectories of autonomous linear system without… 1.1 Front How can we get information about the trajectories of autonomous linear system without…
determining them analytically or using a computer to plot them
1.2 Back Analyzing what happens near critical points....
Captured On [2020-03-16 Mon 19:13] Source Limitations of the Linear: Limit Cycles and Chaos | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the form of the solutions that trace out a closed curve? 1.1 Front What is the form of the solutions that trace out a closed curve?
In a non-linear system
1.2 Back The solution \(\vb{x}(t)\) will be geometrically realized by a point which goes round and round the curve \(C\) with a certain period \(T\)...