Math 2.0

Ordinary differential equations Definition

Fundamental matrix

Builds on System of linear differential equationsFundamental system of solutions

Definition

Let II be an open interval and let A:I→ℝn×nA : I \to \mathbb{R}^{n \times n} be continuous. A fundamental matrix of the homogeneous linear system x′=A(t)xx' = A(t) x is a differentiable function Φ:I→ℝn×n\Phi : I \to \mathbb{R}^{n \times n} with these two properties. For each t∈It \in I, Φ′(t)=A(t) Φ(t)\Phi'(t) = A(t)\, \Phi(t). For each t∈It \in I, the matrix Φ(t)\Phi(t) is invertible.

Remarks

Equivalently, the columns of Φ\Phi are nn solutions that form a basis of the space of solutions. Each solution is x(t)=Φ(t) cx(t) = \Phi(t)\, c for a unique c∈ℝnc \in \mathbb{R}^n, and the solution with x(t0)=x0x(t_0) = x_0 is Φ(t)Φ(t0)−1x0\Phi(t) \Phi(t_0)^{-1} x_0. If Φ′(t)=A(t)Φ(t)\Phi'(t) = A(t) \Phi(t) and Φ(t0)\Phi(t_0) is invertible at one point t0t_0, then Φ(t)\Phi(t) is invertible at each point, by Liouville's formula. For a constant matrix AA, Φ(t)=etA\Phi(t) = e^{tA} is a fundamental matrix. See the matrix exponential as the solution.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionHomogeneous linear equation, definition, no evidence levelHomogeneous linear eq…definitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionSuperposition principle, statement, no evidence levelSuperposition princip…statementSolution space of a homogeneous linear equation has dimension n, statement, no evidence levelSolution space of a h…statementFundamental system of solutions, definition, no evidence levelFundamental system of…definitionFundamental matrix, definition, no evidence levelFundamental matrixdefinition
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Order of a differential equation definition
  6. Linear differential equation of order n definition
  7. Homogeneous linear equation definition
  8. System of linear differential equations definition
  9. Superposition principle statement
  10. Solution space of a homogeneous linear equation has dimension n statement
  11. Fundamental system of solutions definition
  12. Fundamental matrix definition

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6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_matrix_(linear_differential_equation), lead section
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Kinddefinition
WritingW0 stub
MSC34A30
DefinesFundamental matrix
UsesSystem of linear differential equations, Fundamental system of solutions

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