>>13920054As much fun as it is vomiting out words, here's an explanation.
x = position
v = velocity (derivative of position w.r.t time)
a = acceleration (derivative of velocity w.r.t. time)
>differential equationsequations that include both a function and its derivative
Example: 4v^2 + 10x = 7
>second-orderThe equations only include up to the second derivative
Example: 2a^3 + 4v + 10x = 7
>linearNone of the variables (in this case, functions and derivatives) are raised to a power other than one.
Example: 2a + 4v + 10x = 7
>homogeneousThe solution of these equations is always zero, there are no constants.
Example: 2a + 4v + 10x = 0
>why complex numbers?Because these equations are homogeneous, the derivatives have to be a multiple of the original functions, or else they wouldn't add up to zero.
So these equations naturally take the form x = e^(B * t)
This means the derivatives are v = B * e^(B * t) and a = B^2 * e^(B * t)
We can then factor out the exponential term in our earlier equation, and get e^(B * t) [2 * B^2 + 4 * B + 10] = 0
Since exponentials never equal zero, this can be broken down to a quadratic and solved with the quadratic formula. In this example, we get 1 + 2i and 1 - 2i
So x = C * e^([1 + 2i] * t) + D * e^([1 - 2i] * t), where C and D are constants of integration. So we have the equation in terms of imaginary numbers.
But, rather than leaving it like that, we use Euler's Identity to bring it down to x = e^t * [E * cos(2t) + F * sin(2t)], where E and F are constants of integration not equal to C and D.
And voila, we used complex numbers.
>why does any of this matter irlJust look at a mass on a spring.
F = k * x
F = m * a
m * a = k * x
ma - kx = 0
So it's a second-order linear homogeneous differential equation. If you want to know where the mass will be at any time t, you'll need complex numbers.
