University Differential Equations Systems of Linear Differential Equations — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 4 definitions and 8 theorems, 8 worked examples, 1 diagram, in 13 printable pages. This page lists what the notes cover. The notes themselves come with the University Differential Equations bundle.
Unit notes — in the bundle
Not sold separately. The practice worksheet for this unit is free, with no email and no account.
Systems of Linear Differential Equations — what the notes cover
8 sections, in the order the unit is taught. Each entry below is the title of a definition or a theorem the notes state in full.
What you need
Also in this section: 1 note on method and common errors.
Writing a higher-order equation as a first-order system
- Definition — First-order linear system in matrix form
- Theorem — Reducing a higher-order equation to a system
Also in this section: 1 worked example, 1 note on method and common errors.
Matrix form and the general solution
- Definition — Wronskian and the fundamental matrix
- Theorem — Fundamental sets, and existence and uniqueness
Also in this section: 1 worked example, 1 note on method and common errors.
Distinct real eigenvalues
- Definition — Eigenvalues and eigenvectors
- Theorem — Solutions from eigenpairs
Also in this section: 1 worked example, 1 note on method and common errors.
Complex eigenvalues
- Theorem — Real solutions from a complex eigenpair
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
A repeated eigenvalue
- Definition — Generalized eigenvector
- Theorem — A repeated eigenvalue in a system
Also in this section: 1 worked example, 1 note on method and common errors.
Nonhomogeneous systems
- Theorem — Structure of the general solution
- Theorem — Variation of parameters
Also in this section: 2 worked examples, 1 note on method and common errors.
Choosing the form of the solution
- Theorem — Reading the case off the matrix
Also in this section: 1 worked example, 1 note on method and common errors.
Where the notes are
The notes are one of the documents in the University Differential Equations Solutions Bundle, beside the unit test, the challenge problems and every worked answer key. The Systems of Linear Differential Equations practice worksheet is free and stays free.
Every University Differential Equations topic — the complete Solutions Bundle
One download, one payment, the whole program. For all 9 University Differential Equations units: the worksheet, the reference notes, the challenge set, the unit test and every answer key — these notes included.
- Worked solutions, not answer lists — every step written out
- Covers the whole year's program at this level
- Less than the price of one hour of tutoring — for the entire year's solutions
Practise it, then test yourself
Notes on their own do not make anything stick. This unit has a free, printable practice worksheet, and an end-of-unit test that comes with the bundle.
Systems of Linear Differential Equations practice worksheet About the Systems of Linear Differential Equations unit test