Difference between revisions of "Differential Equations"

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Method for solving linear differential equations:
 
Method for solving linear differential equations:
 
# Put equation into standard form: dy/dx + f(x)y = g(x)
 
# Put equation into standard form: dy/dx + f(x)y = g(x)
# Find the integrating factor: u(x) which is equal to e<sup>∫f(x)dx</sup>, so du/dx = f(x)u(x)
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# Find the integrating factor: u(x) which is equal to e<sup>∫f(x)dx</sup>, so du/dx = u(x)f(x)
 
# multiply the standard form by u(x): u(x)dy/dx + u(x)f(x)y = u(x)g(x)
 
# multiply the standard form by u(x): u(x)dy/dx + u(x)f(x)y = u(x)g(x)
# use the product rule (udy/dx +ydu/dx = d/dx(u,y)) on the left side: d/dx(u(x),y(x)) = u(x)g(x)
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# use the product rule (udy/dx + ydu/dx = (u,y)d/dx) on the left side: d/dx(u(x),y(x)) = u(x)g(x)
# integrate both sides
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# integrate both sides, use sources such as [http://integral-table.com/ table of integrals]: u(x)y(x) + C = ∫u(x)g(x)dx
 
# solve for y
 
# solve for y
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 +
==Examples==
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* [[Falling Body with Air Resist]]

Latest revision as of 18:11, 5 May 2020

The integral character ∫

Method for solving linear differential equations:

  1. Put equation into standard form: dy/dx + f(x)y = g(x)
  2. Find the integrating factor: u(x) which is equal to e∫f(x)dx, so du/dx = u(x)f(x)
  3. multiply the standard form by u(x): u(x)dy/dx + u(x)f(x)y = u(x)g(x)
  4. use the product rule (udy/dx + ydu/dx = (u,y)d/dx) on the left side: d/dx(u(x),y(x)) = u(x)g(x)
  5. integrate both sides, use sources such as table of integrals: u(x)y(x) + C = ∫u(x)g(x)dx
  6. solve for y

Examples