Difference between revisions of "Differential Equations"
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# Find the integrating factor: u(x) which is equal to e<sup>∫f(x)dx</sup>, so du/dx = f(x)u(x) | # Find the integrating factor: u(x) which is equal to e<sup>∫f(x)dx</sup>, so du/dx = f(x)u(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 = (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 | # integrate both sides | ||
# solve for y | # solve for y | ||
Revision as of 05:33, 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 = f(x)u(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
- solve for y
