Systems of linear ordinary differential equations use matrices to describe how several dependent variables change together. This cheat sheet covers first order homogeneous systems of the form and how eigenvalues determine solution behavior. It is useful when moving from single differential equations to coupled systems in calculus, differential equations, engineering, and physics.
The goal is to recognize the eigenvalue case quickly and write the correct general solution form.
Key Facts
- A homogeneous linear system is written as , where is a vector of unknown functions and is a constant matrix.
- Eigenvalues are found from the characteristic equation .
- For a real eigenvalue with eigenvector , one solution is .
- If has linearly independent eigenvectors , then .
- For a repeated eigenvalue with one eigenvector and generalized eigenvector satisfying , two solutions are and .
- For complex eigenvalues with eigenvector , real solutions come from and .
- The sign of determines growth or decay: gives decay, gives growth, and gives neutral oscillation in the linear model.
- Initial conditions are applied by substituting into the general solution and solving for the constants .
Vocabulary
- Linear system
- A set of differential equations that can be written in matrix form as for a constant matrix .
- Eigenvalue
- A scalar such that for some nonzero vector .
- Eigenvector
- A nonzero vector that keeps its direction under multiplication by , changing only by the factor .
- Generalized eigenvector
- A vector used for defective repeated eigenvalues, satisfying .
- Fundamental solution set
- A linearly independent collection of solution vectors that can be combined to form every solution of the system.
- Phase portrait
- A geometric picture of solution curves in the state plane showing how the vector evolves over time.
Common Mistakes to Avoid
- Using inconsistently with , which can change signs in intermediate steps. The roots are the same, but algebra errors often appear if the convention is switched mid-problem.
- Forgetting that eigenvectors are vectors, not constants, which leads to writing alone instead of .
- Treating a repeated eigenvalue as automatically giving two independent eigenvectors, which is not always true. If there is only one eigenvector, a generalized eigenvector must be used.
- Leaving complex-valued solutions as the final answer for a real system, which misses the required real solution basis. Use real and imaginary parts to form real solutions.
- Applying the initial condition before forming the full general solution, which can eliminate necessary constants. First build the complete solution, then substitute .
Practice Questions
- 1 Find the eigenvalues and eigenvectors of , then write the general solution to .
- 2 Solve the initial value problem with .
- 3 For , find the complex eigenvalues and describe the real solution behavior.
- 4 Explain how the long-term behavior of solutions changes when all eigenvalues satisfy compared with when at least one eigenvalue satisfies .
Understanding Systems of Linear ODEs with Eigenvalues
The useful idea behind eigenvalues is that they identify the system's natural modes of motion. A mode is a pattern in which all components keep the same fixed ratio while their overall size changes. For example, two connected tanks may have water levels that rise or fall together in one particular ratio.
A mechanical system may have parts moving in a particular shape. Most starting states are mixtures of several modes.
Over time, the mode that grows fastest often becomes the most visible one. This is why a small positive eigenvalue can control long term behavior, even when another mode is shrinking.
For a two variable system, the phase plane gives a picture that formulas alone can hide. Each point represents the current values of the two variables. The system assigns a direction of motion to every point, creating a direction field.
Real eigenvectors point along straight paths that solutions can follow exactly. These paths are important because nearby trajectories may move toward them or away from them. Two negative real eigenvalues usually lead trajectories toward the origin.
Two positive real eigenvalues lead outward. Eigenvalues with opposite signs create a saddle. A saddle is unstable because nearly every nearby starting point eventually moves away, even though a few special paths approach the origin.
Complex eigenvalues describe rotation combined with change in size. The imaginary part sets the turning rate, so a larger imaginary part means more rapid cycles. The real part controls whether the loops move inward or outward.
In a model of a swinging object with friction, inward spirals represent motion that gradually settles down. In an electrical circuit, voltage and current can oscillate while resistance removes energy, producing the same inward pattern. Outward spirals can represent feedback that strengthens a disturbance.
Linear models are often most reliable near an equilibrium point, where the variables are close to a steady state. Far from that point, real systems may have limits that the linear prediction does not include.
Repeated eigenvalues need extra care because the number of independent directions matters more than the repeated number alone. If there are enough independent eigenvectors, the solution has the usual separate mode structure. If there is only one eigenvector, a second solution includes a factor of time.
That factor changes the shape of trajectories and can create shearing behavior. When solving problems, check eigenvectors after finding the eigenvalues. Then use the initial values only after building a complete set of independent solutions.
A good final check is practical. At time zero, the solution must match the given state.
Its derivative at time zero must match the change predicted by the matrix. This check catches sign errors, missing factors of time, and incorrect handling of complex parts.