University Differential Equations Differential Equations and Direction Fields — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 10 definitions and 4 theorems, 6 worked examples, 3 diagrams, 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.
Differential Equations and Direction Fields — what the notes cover
7 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.
Order, linearity and classification
- Definition — Order, and the two variables
- Definition — Linear equation of order
Also in this section: 1 worked example, 1 note on method and common errors.
Solutions, families and initial conditions
- Definition — Solution on an interval
- Definition — General solution, particular solution, initial value problem
- Theorem — Interval of validity
Also in this section: 1 worked example, 1 note on method and common errors.
Direction fields, isoclines and solution curves
- Definition — Direction field and solution curve
- Definition — Isocline
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Autonomous equations and the phase line
- Definition — Autonomous equation and its equilibrium solutions
- Definition — The three kinds of equilibrium
- Theorem — What an autonomous equation forces
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Existence and uniqueness for an initial value problem
- Definition — Normal form
- Theorem — Existence and uniqueness
Also in this section: 1 worked example, 1 note on method and common errors.
Euler's method
- Definition — Euler's method
- Theorem — Concavity fixes the direction of the error
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
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 Differential Equations and Direction Fields 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.
Differential Equations and Direction Fields practice worksheet About the Differential Equations and Direction Fields unit test