lilernoEngineeringOpening Lilerno
Chapter 07 · Guided self-study
Intro + external course studySolve coupled equations and describe a system that changes with time.
Build the idea
An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.
This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.
Start with these prerequisites
Shared science foundations
Machines often have several coupled unknowns. Linear algebra organizes simultaneous equations; differential equations describe how those unknowns change.
Module lens
A matrix keeps several equations, unknowns and units together so a solution can be checked as a whole.
A matrix row is an equation, not just a sequence of numbers. Label the unknown vector and its units. A singular system can have no solution or infinitely many; a numerical answer alone does not prove uniqueness.
A first-order ODE typically needs one initial value. A second-order mechanical ODE needs two, often initial position and velocity. Eigenvalues help describe modes, but their physical meaning depends on the model and sign convention.
See the relationship
Change one quantity, watch the graph respond, then explain the result in your own words.
Predict → change → explain
Move the right-hand sides. Does the intersection still satisfy both equations?
Axes: x (dimensionless) → y (dimensionless). Bounds may rescale when inputs change.
Intersection: x = 2, y = 3. Check by substitution.
Dimensionless system x + y = a and 2x − y = b. Its coefficient matrix is fixed and nonsingular.
Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.
| Series | x (dimensionless) | y (dimensionless) |
|---|---|---|
| x + y = a | -1 | 6 |
| x + y = a | -0.925 | 5.925 |
| x + y = a | -0.85 | 5.85 |
| x + y = a | -0.775 | 5.775 |
| x + y = a | -0.7 | 5.7 |
| x + y = a | -0.625 | 5.625 |
| x + y = a | -0.55 | 5.55 |
| x + y = a | -0.475 | 5.475 |
| x + y = a | -0.4 | 5.4 |
| x + y = a | -0.325 | 5.325 |
| x + y = a | -0.25 | 5.25 |
| x + y = a | -0.175 | 5.175 |
| x + y = a | -0.1 | 5.1 |
| x + y = a | -0.025 | 5.025 |
| x + y = a | 0.05 | 4.95 |
| x + y = a | 0.125 | 4.875 |
| x + y = a | 0.2 | 4.8 |
| x + y = a | 0.275 | 4.725 |
| x + y = a | 0.35 | 4.65 |
| x + y = a | 0.425 | 4.575 |
| x + y = a | 0.5 | 4.5 |
| x + y = a | 0.575 | 4.425 |
| x + y = a | 0.65 | 4.35 |
| x + y = a | 0.725 | 4.275 |
| x + y = a | 0.8 | 4.2 |
| x + y = a | 0.875 | 4.125 |
| x + y = a | 0.95 | 4.05 |
| x + y = a | 1.025 | 3.975 |
| x + y = a | 1.1 | 3.9 |
| x + y = a | 1.175 | 3.825 |
| x + y = a | 1.25 | 3.75 |
| x + y = a | 1.325 | 3.675 |
| x + y = a | 1.4 | 3.6 |
| x + y = a | 1.475 | 3.525 |
| x + y = a | 1.55 | 3.45 |
| x + y = a | 1.625 | 3.375 |
| x + y = a | 1.7 | 3.3 |
| x + y = a | 1.775 | 3.225 |
| x + y = a | 1.85 | 3.15 |
| x + y = a | 1.925 | 3.075 |
| x + y = a | 2 | 3 |
| x + y = a | 2.075 | 2.925 |
| x + y = a | 2.15 | 2.85 |
| x + y = a | 2.225 | 2.775 |
| x + y = a | 2.3 | 2.7 |
| x + y = a | 2.375 | 2.625 |
| x + y = a | 2.45 | 2.55 |
| x + y = a | 2.525 | 2.475 |
| x + y = a | 2.6 | 2.4 |
| x + y = a | 2.675 | 2.325 |
| x + y = a | 2.75 | 2.25 |
| x + y = a | 2.825 | 2.175 |
| x + y = a | 2.9 | 2.1 |
| x + y = a | 2.975 | 2.025 |
| x + y = a | 3.05 | 1.95 |
| x + y = a | 3.125 | 1.875 |
| x + y = a | 3.2 | 1.8 |
| x + y = a | 3.275 | 1.725 |
| x + y = a | 3.35 | 1.65 |
| x + y = a | 3.425 | 1.575 |
| x + y = a | 3.5 | 1.5 |
| x + y = a | 3.575 | 1.425 |
| x + y = a | 3.65 | 1.35 |
| x + y = a | 3.725 | 1.275 |
| x + y = a | 3.8 | 1.2 |
| x + y = a | 3.875 | 1.125 |
| x + y = a | 3.95 | 1.05 |
| x + y = a | 4.025 | 0.975 |
| x + y = a | 4.1 | 0.9 |
| x + y = a | 4.175 | 0.825 |
| x + y = a | 4.25 | 0.75 |
| x + y = a | 4.325 | 0.675 |
| x + y = a | 4.4 | 0.6 |
| x + y = a | 4.475 | 0.525 |
| x + y = a | 4.55 | 0.45 |
| x + y = a | 4.625 | 0.375 |
| x + y = a | 4.7 | 0.3 |
| x + y = a | 4.775 | 0.225 |
| x + y = a | 4.85 | 0.15 |
| x + y = a | 4.925 | 0.075 |
| x + y = a | 5 | 0 |
| 2x − y = b | -1 | -3 |
| 2x − y = b | -0.925 | -2.85 |
| 2x − y = b | -0.85 | -2.7 |
| 2x − y = b | -0.775 | -2.55 |
| 2x − y = b | -0.7 | -2.4 |
| 2x − y = b | -0.625 | -2.25 |
| 2x − y = b | -0.55 | -2.1 |
| 2x − y = b | -0.475 | -1.95 |
| 2x − y = b | -0.4 | -1.8 |
| 2x − y = b | -0.325 | -1.65 |
| 2x − y = b | -0.25 | -1.5 |
| 2x − y = b | -0.175 | -1.35 |
| 2x − y = b | -0.1 | -1.2 |
| 2x − y = b | -0.025 | -1.05 |
| 2x − y = b | 0.05 | -0.9 |
| 2x − y = b | 0.125 | -0.75 |
| 2x − y = b | 0.2 | -0.6 |
| 2x − y = b | 0.275 | -0.45 |
| 2x − y = b | 0.35 | -0.3 |
| 2x − y = b | 0.425 | -0.15 |
| 2x − y = b | 0.5 | 0 |
| 2x − y = b | 0.575 | 0.15 |
| 2x − y = b | 0.65 | 0.3 |
| 2x − y = b | 0.725 | 0.45 |
| 2x − y = b | 0.8 | 0.6 |
| 2x − y = b | 0.875 | 0.75 |
| 2x − y = b | 0.95 | 0.9 |
| 2x − y = b | 1.025 | 1.05 |
| 2x − y = b | 1.1 | 1.2 |
| 2x − y = b | 1.175 | 1.35 |
| 2x − y = b | 1.25 | 1.5 |
| 2x − y = b | 1.325 | 1.65 |
| 2x − y = b | 1.4 | 1.8 |
| 2x − y = b | 1.475 | 1.95 |
| 2x − y = b | 1.55 | 2.1 |
| 2x − y = b | 1.625 | 2.25 |
| 2x − y = b | 1.7 | 2.4 |
| 2x − y = b | 1.775 | 2.55 |
| 2x − y = b | 1.85 | 2.7 |
| 2x − y = b | 1.925 | 2.85 |
| 2x − y = b | 2 | 3 |
| 2x − y = b | 2.075 | 3.15 |
| 2x − y = b | 2.15 | 3.3 |
| 2x − y = b | 2.225 | 3.45 |
| 2x − y = b | 2.3 | 3.6 |
| 2x − y = b | 2.375 | 3.75 |
| 2x − y = b | 2.45 | 3.9 |
| 2x − y = b | 2.525 | 4.05 |
| 2x − y = b | 2.6 | 4.2 |
| 2x − y = b | 2.675 | 4.35 |
| 2x − y = b | 2.75 | 4.5 |
| 2x − y = b | 2.825 | 4.65 |
| 2x − y = b | 2.9 | 4.8 |
| 2x − y = b | 2.975 | 4.95 |
| 2x − y = b | 3.05 | 5.1 |
| 2x − y = b | 3.125 | 5.25 |
| 2x − y = b | 3.2 | 5.4 |
| 2x − y = b | 3.275 | 5.55 |
| 2x − y = b | 3.35 | 5.7 |
| 2x − y = b | 3.425 | 5.85 |
| 2x − y = b | 3.5 | 6 |
| 2x − y = b | 3.575 | 6.15 |
| 2x − y = b | 3.65 | 6.3 |
| 2x − y = b | 3.725 | 6.45 |
| 2x − y = b | 3.8 | 6.6 |
| 2x − y = b | 3.875 | 6.75 |
| 2x − y = b | 3.95 | 6.9 |
| 2x − y = b | 4.025 | 7.05 |
| 2x − y = b | 4.1 | 7.2 |
| 2x − y = b | 4.175 | 7.35 |
| 2x − y = b | 4.25 | 7.5 |
| 2x − y = b | 4.325 | 7.65 |
| 2x − y = b | 4.4 | 7.8 |
| 2x − y = b | 4.475 | 7.95 |
| 2x − y = b | 4.55 | 8.1 |
| 2x − y = b | 4.625 | 8.25 |
| 2x − y = b | 4.7 | 8.4 |
| 2x − y = b | 4.775 | 8.55 |
| 2x − y = b | 4.85 | 8.7 |
| 2x − y = b | 4.925 | 8.85 |
| 2x − y = b | 5 | 9 |
| Solution | 2 | 3 |
Save your place when you finish reading.
Original Lilerno example
Solve x + y = 5 and 2x − y = 1.
Add equations to eliminate y: 3x = 6.
Find x = 2.
Substitute into x + y = 5 to obtain y = 3.
Check both original equations.
Test the model
Open learning, traceable sources
Work in this order. These links open the publisher’s material; free access does not always permit republication.
MIT OpenCourseWare
Start with: Elimination → vector spaces → eigenvalues
Solve small systems by hand before checking with software.
MIT OpenCourseWare
Start with: First-order ODEs → second-order constant-coefficient equations
Check each proposed solution by differentiation and its initial conditions.
Recall, then record
Close the explanation and answer these in your own words. Return tomorrow, then again later in the week.
What does one matrix row mean?
What does singular mean?
Why do initial conditions matter?
Translate a two-unknown force balance into a matrix system and document its units and solution checks.
Self-reported tasks, not an assessment of mastery or university credit. Reading a page does not complete a chapter.
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Open learning, traceable sources
Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.
Elimination, vector spaces, least squares, eigenvalues and matrix models.
CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.
Open source ↗Rights / publisher record ↗First-order equations, second-order oscillators, Laplace transforms and systems. Start with the first-order unit.
CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.
Open source ↗Rights / publisher record ↗