University Differential Equations The Laplace Transform — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 5 definitions and 10 theorems, 7 worked examples, 2 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.
The Laplace Transform — 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.
The transform from the definition and from the pairs
- Definition — The Laplace transform
- Theorem — Existence and decay of a transform
- Theorem — The basic transform pairs
Also in this section: 1 worked example, 1 note on method and common errors.
Inverse transforms and partial fractions
- Definition — The inverse Laplace transform
- Theorem — Inverting a proper rational transform
Also in this section: 1 worked example, 1 note on method and common errors.
Transforms of derivatives and initial value problems
- Theorem — Transform of a derivative, and the method it gives
Also in this section: 1 worked example, 1 note on method and common errors.
Translation on the s-axis and on the t-axis
- Theorem — Translation on the s-axis
- Definition — The unit step function
- Theorem — Translation on the t-axis
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Piecewise and periodic forcing
- Theorem — Transform of a periodic function
- Theorem — The solution at a jump in the forcing
Also in this section: 2 worked examples, 1 note on method and common errors, 1 diagram.
Convolution, impulses and the Dirac delta
- Definition — Convolution
- Theorem — The convolution theorem
- Definition — The Dirac delta function
- Theorem — What an impulse does to a 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 The Laplace Transform 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.
The Laplace Transform practice worksheet About the The Laplace Transform unit test