FirstOrder ODEs

1
First-Order ODEs

• Existence and uniqueness of solutions
• Separable ODEs
• Exact ODEs
• Linear ODEs

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Concept of a Solution

• First-order ODE
• Solution
• Solution must be defined differentiable on an
  interval a lt x lt b
• General solution completely determined up to a
  constant
• Boundary conditions are needed to find constant
  determine complete solution
• Initial value problem

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Existence and Uniqueness of Solutions

• Initial value problem (IVP)
• Three possibilities
• Unique solution
• Infinite number of solutions
• No solution
• Fundamental questions
• Existence under what conditions does the IVP
  have at least one solution?
• Uniqueness under what conditions does the IVP
  have at most one solution?

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Existence Theorem

• Assumptions
• f(x,y) is continuous for
• f(x,y) is bounded in R so
• Existence There exists at least one solution to
  the IVP for

5
Uniqueness Theorem

• Assumptions
• f(x,y) its partial derivative fy(x,y) are
  continuous for
• f(x,y) its partial derivative fy(x,y) are
  bounded in R so
• Uniqueness there exists precisely one solution
  to the IVP for

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Understanding the Theorems

• Necessary sufficient conditions
• If the assumptions are true, then the result is
  true (sufficiency)
• If the assumptions are not true, then the result
  is not true (necessary)
• Lipschitz condition for uniqueness
• Examples
• Non-existence
• Non-uniqueness

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Separable ODEs

• General form
• Separate integrate
• Solve for y(x)
• Evaluate constant

8
Plug-Flow Chemical Reactor
9
Exact ODEs

• First-order ODE
• Exactness condition implicit solution form
• Solution equation
• Constant of integration

10
Integrating Factors

• Attempt to create exact equation
• Finding the integration factor F(x)
• Check condition
• Integrating factor

11
Linear First-Order ODEs

• General form
• Homogeneous solution r(x) 0
• Non-homogeneous solution

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Bernoulli Equation

• General form
• Linear only if a 0 or a 1
• Variable transformation
• Linear transformed equation

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Example

• Transform ODE
• Solve transformed ODE
• Integrate by parts

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Example cont.

• Substitute result
• Transform solution
• Evaluate constant c using initial condition y(x0)