← Mathematics — Year 2/Ordinary Differential Equations: Methods and Systems

Linear Systems of ODEs

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Linear Systems of ODEs

For $x'=Ax$ , eigenvalues and eigenvectors generate modal solutions $e^{\lambda t}v$ . The spectrum of $A$ determines growth, decay, oscillation and stability. The distinction between a formal hypothesis and an intuitive picture is made explicit so that calculations can be justified, not merely patterned.

Worked reasoning

Each modal exponential decays.
All modes tend to zero, giving asymptotic stability.

Key idea

The spectrum of $A$ determines growth, decay, oscillation and stability.

If all eigenvalues of $A$ have negative real parts, the origin of $x'=Ax$ is typically:

Asymptotically stable

Unstable

A centre only

Undefined

Which statement best captures the central mathematical idea in Linear Systems of ODEs?

The spectrum of $A$ determines growth, decay, oscillation and stability.

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Linear Systems of ODEs, which move is most reliable?

Write the governing definition and list its hypotheses before calculating in linear systems of odes.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Linear Systems of ODEs?

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

The spectrum of $A$ determines growth, decay, oscillation and stability.

Write the governing definition and list its hypotheses before calculating in linear systems of odes.

Test a representative example and a boundary or counterexample, then verify the conclusion directly from the definition.

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