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Definition

A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.

Differential Equation of n-th Order

$$a_n y^{(n)} + ... + a_1 y' + a_0 y = b_m x^{(m)} + ... + b_1 x' + b_0 x$$

with the unknown function $y$ and its $n$ derivates $y',y'',...$, the known function $x$ and the coefficients $a_i, b_j$

DGL-Systeme

Jede DGL lässt sich reduzieren auf ein DGL-System 1. Ordnung:

  1. Substituiere $x_i := y^{(i-1)}$ und drücke $\dot x_i$ durch $x_1,...,x_n$ aus.
    ${\Rightarrow}$ mit ${\boldsymbol x}_{{\rm}ges} = {\boldsymbol x}_{{\rm}hom} + {\boldsymbol x}_{{\rm}part}$
    Hom. Lösung: 1. Bestimme EW $\lambda_i$ und Basis aus EV ${\boldsymbol b}_i$ von ${\boldsymbol{A}}$

  2. ${\boldsymbol x}_{{\rm}hom} = {\boldsymbol c} \cdot e^{(x-x_0){\boldsymbol{A}}} = \sum\limits_{i = 0}^n c_i \cdot e^{\lambda_i x} \cdot {\boldsymbol b}_i$

  3. Bestimmung der Konstanten durch einsetzen der Anfangsbedingungen!

Lösen von homogenen DGLs 2. Ordnung

Gegeben: Homogene Differnetialgleichungen der Form ${\boldsymbol \dot x} = {\boldsymbol{A}} {\boldsymbol x}$ mit Anfangswerten $x_{0,1}$ und $x_{0,2}$

Depending on the Eigenvalues $\lambda_1, \lambda_2 \in {\mathop{\mathbb R}}$

1. If Eigenvalues are real and inequal: $\Large \lambda_1 \ne \lambda_2$

$$\bigl| \lambda_1 \bigr| < \bigl|\lambda_2 \bigr| {\Rightarrow}{\boldsymbol q}_2 \text{,,schneller''} $$

$${\boldsymbol x}(t) = x_{0,1} \cdot \exp(\lambda_1 t) \cdot {\boldsymbol q}_1 + x_{0,2} \cdot \exp(\lambda_2 t) \cdot {\boldsymbol q}_2$$

Matrix $\Lambda$ Eigenwerte ${\boldsymbol x} = 0$ Name Portrait
${\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}}$ $\lambda_1 \lt 0 \lt \lambda_2$ instable Sattelpunkt
$\lambda_2 \lt 0, \lambda_1 \lt 0$ stable Knoten 2
$0 \lt \lambda_1, 0 \lt \lambda_2$ instable Knoten 2
${\begin{bmatrix} 0 & 0 \\ 0 & \lambda_2 \end{bmatrix}}$ $\lambda_1 = 0,\ \lambda_2 \lt 0$ stabil Kamm
$\lambda_1 = 0,\ \lambda_2 \gt 0$ instable Kamm

2. If Eigenvalues are real and equal: $\Large \lambda_1 = \lambda_2$

$${\boldsymbol{Q}}' = {\begin{bmatrix} {\boldsymbol q}'_1 & {\boldsymbol q}'_2 \end{bmatrix}} = {\begin{bmatrix} -a_{12} & -a_{12} \\ \frac{a_{11} - a_{22}}{2} & \frac{a_{11} - a_{22}}{2} -1 \end{bmatrix}} = {\begin{bmatrix} {\begin{pmatrix} \text{Eigen-} \\ \text{vektor} \end{pmatrix}} & {\begin{pmatrix} \text{Haupt-} \\ \text{vektor} \end{pmatrix}} \end{bmatrix}}$$

$${\boldsymbol x}(t) = \left[ {\boldsymbol{1}} + ({\boldsymbol{A}} - \lambda {\boldsymbol{1}}) \cdot t \right] \cdot \exp(\lambda t) \cdot {\begin{pmatrix} x_{0,1} \\ x_{0,2} \end{pmatrix}}$$

Matrix $\Lambda$ Eigenwerte ${\boldsymbol x} = 0$ Name Portrait
${\begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix}}$ $\lambda \lt 0$ stabil Knoten 1
$\lambda \gt 0$ instable Knoten 1
${\begin{bmatrix} \lambda & 1 \\ 0 & \lambda \end{bmatrix}}$ $\lambda \lt 0$ stabil Knoten 3
$\lambda \gt 0$ instable Knoten 3
${\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}}$ $\lambda = 0$ stabil Ruheebene
${\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}}$ $\lambda = 0$ instable Ruhegerade }

3. If Eigenvalues are complex and equal: $\Large {\boldsymbol{\lambda}}_1 = {\boldsymbol{\lambda}}^*_2$

$${\boldsymbol{\lambda}}_1 = {\boldsymbol{\lambda}}^*_2 = \alpha + \beta j \in {\mathop{\mathbb C}}$$

$${\boldsymbol{Q}}' = {\begin{bmatrix} \Re{{\boldsymbol q}_1} & \Im{{\boldsymbol q}_1} \end{bmatrix}} = {\begin{bmatrix} {\boldsymbol q}_r & {\boldsymbol q}_j \end{bmatrix}}$$

$$\begin{array}{rl} {\boldsymbol x}(t) & = x_{0,1} \cdot e^{\alpha t} \cdot \left[ \cos (\beta t) {\boldsymbol q}_r - \sin(\beta t) \cdot {\boldsymbol q}_j \right] + \\ & +\, x_{0,2} \cdot e^{\alpha t} \cdot \left[ \sin (\beta t) {\boldsymbol q}_r + \cos(\beta t) \cdot {\boldsymbol q}_j \right] \end{array}$$

Matrix $\Lambda$ Eigenwerte ${\boldsymbol x} = 0$ Name Portrait
${\begin{bmatrix} \alpha & -\beta \\ \beta & \alpha \end{bmatrix}}$ $\alpha \lt 0,\ \beta \ne 0$ stabil Strudel
$\alpha \gt 0,\ \beta \ne 0$ instable Strudel
${\begin{bmatrix} 0 & -\beta \\ \beta & 0 \end{bmatrix}}$ $\alpha = 0,\ \beta \ne 0$ stabil Wirbel

Zeitverlauf immer von ${\boldsymbol q}_j$ nach ${\boldsymbol q}_r$ bzw. von ${\boldsymbol q}_r$ nach $-{\boldsymbol q}_j$

Lösung für inhomogene DGL

Inhomogene DGL (${\boldsymbol v} \ne 0$) mit singulärer Matrix ${\boldsymbol{A}}$ (nicht entkoppelbar):

Matrix $\Lambda$ Eigenwerte ${\boldsymbol x} = 0$ Name Portrait
${\begin{bmatrix} 0 & 0 \\ 0 & \lambda_2 \end{bmatrix}}$ $\lambda_1 = 0, \lambda_2 \lt 0$ instable Kamm
${\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}}$ $\lambda = 0$ instable Knoten
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