Math 2.0

Ordinary differential equations Definition

Fundamental system of solutions

Builds on Homogeneous linear equationSolution space of a homogeneous linear equation has dimension n

Definition

Let II be an open interval and let a0,…,an−1:I→ℝa_0, \dots, a_{n-1} : I \to \mathbb{R} be continuous. Let SS be the vector space of solutions on II of the homogeneous linear equation y(n)+an−1(x)y(n−1)+⋯+a0(x)y=0y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0. A fundamental system of solutions of the equation is a basis (y1,…,yn)(y_1, \dots, y_n) of SS.

Remarks

Since SS has dimension n, a family of nn solutions is a fundamental system if and only if it is linearly independent. The Wronskian test checks this. Each solution is then c1y1+⋯+cnync_1 y_1 + \dots + c_n y_n for unique constants c1,…,cnc_1, \dots, c_n. Example: (cos⁡x,sin⁡x)(\cos x, \sin x) is a fundamental system of y′′+y=0y'' + y = 0 on ℝ\mathbb{R}.

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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…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…definition
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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. Superposition principle statement
  9. Solution space of a homogeneous linear equation has dimension n statement
  10. Fundamental system of solutions definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_differential_equation, Linear differential operator
About this page
Tag0H4KNNCHPY
URIm2://0H4KNNCHPY
Aliasesfundamental-set-of-solutions
Kinddefinition
WritingW1 stated
MSC34A30
DefinesFundamental system of solutions
UsesHomogeneous linear equation, Solution space of a homogeneous linear equation has dimension n

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