University Differential Equations Homogeneous Linear Equations with Constant Coefficients — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 5 definitions and 8 theorems, 7 worked examples, 1 diagram, in 14 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.
Homogeneous Linear Equations with Constant Coefficients — 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.
Complex numbers in rectangular, polar and exponential form
- Definition — Rectangular form, conjugate and modulus
- Definition — Polar and exponential form
- Theorem — Euler's formula and arithmetic in exponential form
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Powers and roots of a complex number
- Theorem — Powers in exponential form
- Definition — The th roots of a complex number
Also in this section: 1 worked example, 1 note on method and common errors.
Superposition, linear independence and the Wronskian
- Theorem — Superposition
- Definition — Linear independence and the Wronskian
- Theorem — Fundamental set and the general solution
Also in this section: 1 worked example, 1 note on method and common errors.
Distinct real roots of the characteristic equation
- Definition — The characteristic equation
- Theorem — Distinct real roots
Also in this section: 1 worked example, 1 note on method and common errors.
Repeated roots of the characteristic equation
- Theorem — A repeated real root
Also in this section: 1 worked example, 1 note on method and common errors.
Complex conjugate roots of the characteristic equation
- Theorem — A conjugate pair of complex roots
Also in this section: 1 worked example, 1 note on method and common errors.
Homogeneous equations of third and higher order
- Theorem — From the root list to the general solution
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 Homogeneous Linear Equations with Constant Coefficients 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.
Homogeneous Linear Equations with Constant Coefficients practice worksheet About the Homogeneous Linear Equations with Constant Coefficients unit test