Zeigen Sie: Die homogene lineare Differentialgleichung 3. Ordnung $$ y^{\prime \prime \prime}-3 y^{\prime \prime}+9 y^{\prime}+13 y=0 $$ besitzt die linear unabhängigen Lösungen (Basisfunktionen) \(y_{1}=\mathrm{e}^{-x}\). \(y_{2}=\mathrm{e}^{2 x} \cdot \sin (3 x)\) und \(y_{3}=\mathrm{e}^{2 x} \cdot \cos (3 x)\) Wie lautet die allgemeine Lösung dieser Differentialgleichung?

A homogeneous linear differential equation is a type of equation that relates a function to its derivatives. It has the general form: \[a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_1(x)y' + a_0(x)y = 0\] In this equation, all terms involve the function or its derivatives, and everything is equal to zero.
For our problem, the third-order differential equation is: \[y''' - 3y'' + 9y' + 13y = 0\] This means it relates the third, second, and first derivatives of function y with certain coefficients and sums up to zero.
Homogeneous equations typically have solutions based on exponential, sine, and cosine functions. Finding these solutions involves checking which functions satisfy the equation when substituted in.

Basis functions (Basisfunktionen) are essential in solving differential equations. They are linearly independent solutions that form the foundation to express the general solution.
In our problem, we are given three basis functions: \(y_1 = e^{-x} \, y_2 = e^{2x} \sin(3x) \und y_3 = e^{2x} \cos(3x)\).
To confirm that these are indeed solutions, follow these steps:
1. Plug each function into the differential equation to check if it satisfies the equation. For instance, substitute \(y_1 = e^{-x}\) and compute derivatives up to the third order, then substitute into \(y''' - 3y'' + 9y' + 13y = 0\). Similarly, repeat the process for \(y_2\) and \(y_3\).
2. After verifying each, you use them to build the general solution of the differential equation by combining with constants.

The Wronskian is a determinant used to test the linear independence of a set of functions. For functions to be linearly independent, their Wronskian (a specific determinant) must not be zero.
For our functions \(y_1 = e^{-x}, y_2 = e^{2x} \sin(3x), \ und \ y_3 = e^{2x} \cos(3x}\), we need to set up the Wronskian.
The Wronskian \(W(y_1, y_2, y_3)\) is given by the determinant of the matrix formed by these functions and their first two derivatives:
\ \[ W(y_1, y_2, y_3) = \begin{vmatrix} y_1 & y_2 & y_3 \ y_1' & y_2' & y_3' \ y_1'' & y_2'' & y_3'' \end{vmatrix}\]
After calculating this determinant and confirming it is non-zero, we establish that \(y_1, y_2\ und \ y_3\) are linearly independent.
Combining these independent solutions, the general solution of the differential equation is: \[ y = C_1 e^{-x} + C_2 e^{2x} \sin(3x) + C_3 e^{2x} \cos(3x) \].