math:calculus [AE Wiki]

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====== Calculus ======

"calculus of infinitesimals", study of continous change

===== Differential calculus =====

study of rates ath which quantities change.

==== Derivative ====

Example of a straigth line $y=mx+b$, $m=\Delta y / \Delta x$

"Differentialquotient":

$m = \lim_{x_1 -> x_0} \frac{f(x_1) - f(x_0)}{x_1 - x_0}$

$f'(x) = \frac{d}{dx} f(x) = \lim_{h -> 0} \frac{f(x + h) - f(x)}{h}$

$f''(x) = \frac{d^2}{dx^2} f(x)$

Geometrically, derivative of f at point x=a is slope of tangent line. 

==== Differential equations ====

Relation between collection of functions and their derivatives.

Ordinary differential equation:

Differential operator

$f'(x)=f(x)^2 * x$

$\frac{dy}{dx} = y^2 * x$

$dy = y^2 * x {dx}$

$\frac{1}{y^2} dy = x {dx}$

$\int \frac{1}{y^2} dy = \int x {dx}$

$-y^{-1} + c_1 = 1/2 x^2 + c_2$

$-(1/y)  = 1/2 x^2 + c_2 - c_1$ $(c_2 - c_1 = c)$

$y = \frac{-1}{(1/2 x^2 + c)}$ (y is a function not value, obviously)

=== Types ===

$y' + f(x) y = g(x)$ first order, linear, "homogen" if g(x) = 0, else "inhomogen" ("Störfunktion")

Solve inhomogen eqs:

$y'' + a_1 y' + a_0 * y = g(x)$

$y(x) = y_a(x) + y_p(x)$

General solution of homogen DE $y_a(x)$ (set to 0, solve with characteristic polynom)
"Particular" solution of inhomogen DE $y_p(x)$

v=6jypcZkrvLM
Mit Störfkt, charakteristisches Polynom, lösen.