[QUOTE="Blood-Scribe"][QUOTE="ghoklebutter"] I've studied Laplace transforms a bit - they're absolutely lovely! Also, I may be voicing a minority opinion here, but I truly adore the annihilator method (the one that involves "factoring out" the dependent variable out of the derivatives of a constant coefficient DE to find a characteristic polynomial consisting of differential operators).ghoklebutter
We never really went over the annihilator method in depth, sadly. The professor just mentioned it off-hand and showed us the notation and that was pretty much it. Didn't even do an example with it :< Although it sounds a lot like undetermined coefficients (which was balls and completely inferior to variation of parameters).It's pretty nice actually - here's a typical example:
y'' + 5y' + 6y = e^x
[D^2]y + 5Dy + 6y = e^x
[D^2 + 5D + 6](y) = e^x
Because [D - 1]e^x = De^x - e^x = e^x - e^x = 0
Multiply each side by [D - 1] to annihilate the right side of the DE:
[D - 1][D^2 + 5D + 6](y) = [D - 1][D + 2][D + 3](y) = 0
D = -3, -2, 1, where D = 1 is a root for the nonhomogenous solution and D = -3, -2 are roots for the homogenous one.
The complete solution is y(x) = ae^(-3x) + be^(-2x) + ce^x
The only real downside is that it only works for nonhomogenous equations that involve exponential, polynomial, and/or trigonometric functions on the right side.
Okay, so basically you just treat the differential operators as factors and then multiply the initial equation by whatever it is that annihilates the opposite side and solve for the roots. That's actually pretty neat.e: Would've been nice to have known that instead of having to guess with undeterminied coefficients ;[
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