Difference between revisions of "Differential Equations"
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# 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 = (u,y)d/dx) on the left side: d/dx(u(x),y(x)) = u(x)g(x) | # 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, use sources such as [http://integral-table.com/ table of integrals]: u(x)y(x) = ∫u(x)g(x)dx | + | # 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 | ||
==Examples== | ==Examples== | ||
* [[Falling Body with Air Resist]] | * [[Falling Body with Air Resist]] | ||
Latest revision as of 18:11, 5 May 2020
The integral character ∫
Method for solving linear differential equations:
- Put equation into standard form: dy/dx + f(x)y = g(x)
- Find the integrating factor: u(x) which is equal to e∫f(x)dx, 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)
- 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, use sources such as table of integrals: u(x)y(x) + C = ∫u(x)g(x)dx
- solve for y
