Math 2.0

Ordinary differential equations Definition

Singular solution

Builds on Solution of an ordinary differential equationInitial value problemGeneral solution and particular solution

Definition

Let F(x,y,y′)=0F(x, y, y') = 0 be a first-order ordinary differential equation, and let ys:I→ℝy_s : I \to \mathbb{R} be a solution on an open interval II. The solution ysy_s is singular if uniqueness fails at each point of its graph. This means the following. For each x0∈Ix_0 \in I, there is another solution yy on an open interval containing x0x_0 with y(x0)=ys(x0)y(x_0) = y_s(x_0) and y′(x0)=ys′(x0)y'(x_0) = y_s'(x_0). This solution differs from ysy_s on each neighbourhood of x0x_0.

Remarks

Some authors use a weaker meaning: a solution at which the initial value problem fails to have a unique solution at some point. A singular solution is often the envelope of a family of solutions. It is not a member of the general solution for any value of the constants. Example: for (y′)2=4y(y')^2 = 4 y, the solutions y=(x−C)2y = (x - C)^2 are tangent to the solution y=0y = 0, so y=0y = 0 is singular. Clairaut's equation gives other examples. By the uniqueness theorem, an equation y′=f(x,y)y' = f(x, y) with ff Lipschitz in yy has no singular 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…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionGeneral solution and particular solution, concept, no evidence levelGeneral solution and …conceptInitial value problem, definition, no evidence levelInitial value problemdefinitionSingular solution, definition, no evidence levelSingular solutiondefinition
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  1. Differential equation concept
  2. Function concept
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  5. Solution of an ordinary differential equation definition
  6. General solution and particular solution concept
  7. Initial value problem definition
  8. Singular solution definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Singular_solution, lead section
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DefinesSingular solution
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