Linear DE Theory

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Linear DE Theory

• MATH 224

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Classifications

• by order
• by linearity
• by homongeous or non-homogeneous

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Examples of Differential Equations
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More examples
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Notes
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Focus on Linear DEs

• Lots of useful general theory exists for linear
  DEs
• very little for general non-linear DEs
• Many non-linear problems can be approximated by
  linear DEs
• For full non-linear problems, numerical solvers
  used most in practice
• some theoretical work on interesting classes of
  problems

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Example
Show that the following two functions are
solutions
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Principle of Superposition

• If y1 and y2 are solutions to a homogeneous
  linear DE, then any linear combination of y1 and
  y2 is also a solution
• true also for y1, yn for any linear DE

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In context of previous example
Knowing that two solutions are
other solutions could be
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Fundamental Set of Solutions

• For an n-th order linear, homogeneous DE,
• any set y1, y2, , yn of linearly independent
  solutions on an interval I is called a
  fundamental set of solutions
• The general solution to such a DE is the general
  lin. combination of that set

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Linear Independence

• A set of functions/vectors is linearly
  independent if one cannot be written as a linear
  combination of the others
• Examples

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Further examples
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Back to Example
Knowing that two solutions are
all solutions will be of the form
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Mathematical goal Homogeneous DEs

• If we have a homogeneous linear DE, we want to
• find n linearly independent solutions (preferably
  simplest possible)
• create the general solution as a linear
  combination of those solutions

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Non-homogeneous
The non-homogeneous part changes the solution.
We can find through experimentation that one
solution to this new DE is
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Nomenclature I

• We call yp -2x a particular solution for the
  non-homogeneous DE
• part of solution that is needed to satisfy
  non-homogeneous part of the DE
• We call yc c1/x c2 x4 the complementary
  function, and it is the solution to the
  corresponding homogeneous DE

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Mathematical goal Homogeneous DEs

• If we have a non-homogeneous linear DE, we want
  to
• find n linearly independent solutions to the
  corresponding homogeneous DE, y1, y2, , yn
• find a particular solution to the non-homogenous
  DE, yp
• create the general solution as

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Check for example
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Modeling Example Spring/Mass System

• Mass attached to a spring, with drag and external
  force applied
• Goal predict (quantitatively) the position of
  mass over time

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Diagram
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Newton's 2nd Law
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Continued
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Theory in Practice

• DE is linear, so
• There will be 2 linearly independent solutions to
  the homogeneous form of DE
• There will be a particular solution, yp,
• All possible motions of the spring will be
  predicted by the combination of these
  mathematical formulas

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Reading

• Sections 4.1, 5.1