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--- math:calculus [2019/12/22 21:32] – [Differential equations] phreazer
+++ math:calculus [2019/12/22 22:28] (current) – [Differential equations] phreazer
@@ Line 10: / Line 10: @@
 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.
@@ Line 19: / Line 27: @@
 Ordinary differential equation:
-$y' = 2y + x^2$ = $f'(x)=2*f(x)+x^2$ = $\frac{dy}{dx}$
+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.