The exercises have been developed on foundation of Khan-Exercise Framework.

They offer you the opportunity to engage interactively, individually ("Hints") and repeatedly ("New Numbers") with a mathematical topic. Try to get the result right two or three times in a row. There's no record, thus you are free to try. This is valid for the entire course.

Please note: We are currently in the process of translating the exercises from German into English, powered by AI tools. Once this is done, proofreading is still required. If you still spot typos or similar, you are welcome to collect them and report them in class in January.

1. Ordinary Differential Equations (ODE)
2. (Foundation of) Linear Algebra
3. (Homogeneous) Linear ODE-Systems \(y' =Ay\)
4. Non Linear Systems \(y' = F(y), F: \mathbb R^2 \to \mathbb R^2 \)
5. Euclidean Spaces and Fourier series
6. Miscellaneous

Get acquainted, finding particular solution

Linear Inhomogenity, 2nd order

Stationary Solutions \( y_\infty \) for \( y' = F(y) \)

Finding stationary solution, Stationary solution: convergence

Applications Slope Field

Getting ODE with slope field, Getting ODE again with slope field, Getting ODE with slope field: quadratic

Reading slope field, Reading slope field: quadratic, Reading slope field: quadratic (not normalised)

Finding convergence with slope field (MC), Finding image with slope field (MC)

Linear ODE with constant coefficients

Finding stationary solution, Finding general solution

Finding particular solution, Finding particular solution, Finding values of particular solution

(Non) homogeneous ODE (Integrating Factors)

Finding values of particular solution, Finding more values of particular solution, Finding even more values of particular solution, ... and again

Finding particular solution

Finding values after separation of variables

Getting solutions with separation

Getting more solutions with separation

even more

and more

Getting general solution in the real case, Getting general solution in the complex case

Getting values of particular solution

Matrix-Vector- and Matrix-Matrix-Product

Matrix-Vector-Product \( A \cdot v\), Matrix-Matrix-Product \( A \cdot B\)

Dimension Matrix-Matrix-Product \( A \cdot B\)

Linear Maps

Getting Matrix-map \(v \mapsto A \cdot v\), Getting another Matrix-map \(v \mapsto A \cdot v\)

Values for linear \(\mathcal F : V \to \mathbb R\), More values for linear \(\mathcal F : V \to \mathbb R\)

Eigenvalues and vectors

Eigenvalues of \( 2 \times 2\) - Matrix, Getting \( 2 \times 2\) - Matrix with given EVal, Getting \( 4 \times 4\) - Matrix with given EVal

Finding EVec, Getting \( 2 \times 2\) - Matrix with given EVec, Getting \( 3 \times 3\) - Matrix with given EVec

Getting EVal and EVec

Matrix-Vector-convergence

Determinant

Inverse of \( 2 \times 2 \) - Matrix

Computing \(\det \) of \( 3 \times 3 \) - Matrix

Getting solution of homogeneous linear system with nontrivial solution

Gaussian Algorithms

System in triangle form, Row echelon form as triangle, One step to get row echelon form as triangle

Gauss \( 3 \times 3 \) LGS: unique

Miscellaneous

Computing Coordinate vector

Determine Jordan-form

Existence of stationary solution, Getting stationary solution

Getting system with given stationary solution

Convergence

Getting stationary solution for nonhomogeneous case

Matrix-Exponential \(e^A\)

Matrix exponential \(e^A\) for \( 2 \times 2\) - Matrix \(A\)

Plane Curves

Straight line, Straight line: slower, Straight line: faster

Linearisation and Taylor polynomial

\(T_2(x) \), degree 2, \(T_3(x) \), degree 3

Vector fields

Getting values geometrically, Identifying vector field (MC)

Fixed Points of \(y' = F(y), F: \mathbb R^2 \to \mathbb R^2 \)

Finding fixed points, Finding more fixed points

Finding stable fixed points

Periods

General 1, General 2

Partial Integration

Exercise 1, Exercise 2, Exercise 3

Fourier coefficients

Fourier coefficients of the derivative

Applying Symmetry

Even 1, Even 2, Odd 1, Odd 2

Scalar products

Calculation, More Calculation

Scalar products in \(C^0([a,b], \mathbb R)\)

Scalar products in \(\mathcal P_{\leq n}\) and Orthogonal in \(\mathcal P_{\leq n}\)

Coordinates revisited

Coordinates with scalar product

Lengths

Lengths in \(\mathcal P_{\leq n}\), More lengths in \(\mathcal P_{\leq n}\)

Complex numbers

Arithmetic with Complex Numbers

Sum, Product, Quotient

Multiplication in polar form, Division in polar form

Geometry with Complex Numbers

Given Cartesian coordinates

Addition geometrically, Subtraction geometrically

Finding polar form

Approximating product

Trigonometric Functions

Shift for sine

Frequency modulation

Addition theorem, Another Addition theorem

Solving equation

Polynomial Division

Degree 3 divided by linear , Degree 4 divided by quadratic, Cubic zeroes