Differential Equation#

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:

Substituiere \(x_i := y^{(i-1)}\) und drücke \(\dot x_i\) durch \(x_1,...,x_n\) aus.
\(\Ra\) \boxed{ \dot{\vec x}(t) = \ma A \vec x(t) + \vec s(t) } mit \(\vec x_{\ir ges} = \vec x_{\ir hom} + \vec x_{\ir part}\)
Hom. Lösung: 1. Bestimme EW \(\lambda_i\) und Basis aus EV \(\vec b_i\) von \(\ma A\)
\(\vec x_{\ir hom} = \vec c \cdot e^{(x-x_0)\ma A} = \sum\limits_{i = 0}^n c_i \cdot e^{\lambda_i x} \cdot \vec b_i\)
Bestimmung der Konstanten durch einsetzen der Anfangsbedingungen!

Lösen von homogenen DGLs 2. Ordnung#

Gegeben: Homogene Differnetialgleichungen der Form \(\vec{\dot x} = \ma A \vec x\) mit Anfangswerten \(x_{0,1}\) und \(x_{0,2}\)

Depending on the Eigenvalues \(\lambda_1, \lambda_2 \in \R\)

1. If Eigenvalues are real and inequal: \(\Large \lambda_1 \ne \lambda_2\)#

\[\bigl| \lambda_1 \bigr| < \bigl|\lambda_2 \bigr| \Ra \vec q_2 \text{,,schneller''} \]

\[\vec x(t) = x_{0,1} \cdot \exp(\lambda_1 t) \cdot \vec q_1 + x_{0,2} \cdot \exp(\lambda_2 t) \cdot \vec q_2\]

Matrix \(\Lambda\)	Eigenwerte	\(\vec x = 0\)	Name
\(\mat{\lambda_1 & 0 \\ 0 & \lambda_2}\)	\(\lambda_1 < 0 < \lambda_2\)	instable	Sattelpunkt
	\(\lambda_2 < 0, \lambda_1 < 0\)	stable	Knoten 2
	\(0 < \lambda_1, 0 < \lambda_2\)	instable	Knoten 2
\(\mat{0 & 0 \\ 0 & \lambda_2}\)	\(\lambda_1 = 0,\ \lambda_2 < 0\)	stabil	Kamm
	\(\lambda_1 = 0,\ \lambda_2 > 0\)	instable	Kamm

2. If Eigenvalues are real and equal: \(\Large \lambda_1 = \lambda_2\)#

\[\ma Q' = \mat{ \vec q'_1 & \vec q'_2} = \mat{ -a_{12} & -a_{12} \\ \frac{a_{11} - a_{22}}{2} & \frac{a_{11} - a_{22}}{2} -1 } = \mat{ \vect{\text{Eigen-} \\ \text{vektor}} & \vect{\text{Haupt-} \\ \text{vektor}}}\]

\[\vec x(t) = \left[ \ma 1 + (\ma A - \lambda \ma 1) \cdot t \right] \cdot \exp(\lambda t) \cdot \vect{x_{0,1} \\ x_{0,2}}\]

Matrix \(\Lambda\)	Eigenwerte	\(\vec x = 0\)	Name	Portrait
\(\mat{\lambda & 0 \\ 0 & \lambda}\)	\(\lambda < 0\)	stabil	Knoten 1
	\(\lambda > 0\)	instable	Knoten 1
\(\mat{\lambda & 1 \\ 0 & \lambda}\)	\(\lambda < 0\)	stabil	Knoten 3
	\(\lambda > 0\)	instable	Knoten 3
\(\mat{0 & 0 \\ 0 & 0 }\)	\(\lambda = 0\)	stabil	Ruheebene
\(\mat{0 & 1 \\ 0 & 0 }\)	\(\lambda = 0\)	instable	Ruhegerade	}

3. If Eigenvalues are complex and equal: \(\Large \cx \lambda_1 = \cx \lambda^*_2\)#

\[\cx \lambda_1 = \cx \lambda^*_2 = \alpha + \beta j \in \C\]

\[\ma Q' = \mat{\Re{\vec q_1} & \Im{\vec q_1}} = \mat{ \vec q_r & \vec q_j}\]

\[\begin{array}{rl} \vec x(t) & = x_{0,1} \cdot e^{\alpha t} \cdot \left[ \cos (\beta t) \vec q_r - \sin(\beta t) \cdot \vec q_j \right] + \\ & +\, x_{0,2} \cdot e^{\alpha t} \cdot \left[ \sin (\beta t) \vec q_r + \cos(\beta t) \cdot \vec q_j \right] \end{array}\]

Matrix \(\Lambda\)	Eigenwerte	\(\vec x = 0\)	Name
\(\mat{\alpha & -\beta \\ \beta & \alpha}\)	\(\alpha < 0,\ \beta \ne 0\)	stabil	Strudel
	\(\alpha > 0,\ \beta \ne 0\)	instable	Strudel
\(\mat{0 & -\beta \\ \beta & 0}\)	\(\alpha = 0,\ \beta \ne 0\)	stabil	Wirbel

Zeitverlauf immer von \(\vec q_j\) nach \(\vec q_r\) bzw. von \(\vec q_r\) nach \(-\vec q_j\)

Lösung für inhomogene DGL#

Inhomogene DGL (\(\vec v \ne 0\)) mit singulärer Matrix \(\ma A\) (nicht entkoppelbar):

Matrix \(\Lambda\)	Eigenwerte	\(\vec x = 0\)	Name	Portrait
\(\mat{0 & 0 \\ 0 & \lambda_2}\)	\(\lambda_1 = 0, \lambda_2 < 0\)	instable	Kamm
\(\mat{0 & 1 \\ 0 & 0 }\)	\(\lambda = 0\)	instable	Knoten