Mathe_III_BI

Table of Contents

• Foreword . . . . i

1. Differential Equations . . . . 11.1. Ordinary Differential Equations . . . . 31.1.1. Notation . . . . 31.1.2. Fundamental Definitions . . . . 31.1.3. Existence and Uniqueness of Solutions . . . . 51.1.4. Differential Equations with Separated Variables . . . . 101.1.5. Linear Differential Equations of First Order . . . . 141.1.6. Bernoulli Differential Equations . . . . 161.1.7. Exact Differential Equations . . . . 181.1.8. 2nd Order DE: Right Side does not depend on u . . . . 231.1.9. (∗) 2nd Order DE: Right Side does not explicitly depend on x . . . . 251.1.10. Linear 2nd Order DE with Constant Coefficients . . . . 301.1.11. Linear 2nd Order DE with Variable Coefficients . . . . 401.2. Systems of Differential Equations . . . . 471.2.1. Notation . . . . 471.2.2. Linear Systems with Constant Coefficients . . . . 471.2.3. Higher Order Linear DE as System . . . . 511.3. Numerical Methods for Solutions . . . . 541.3.1. Some Important Methods . . . . 551.3.2. Convergence of Single-step Methods . . . . 581.3.3. Stiff Differential Equations . . . . 611.4. Partial Differential Equations . . . . 661.4.1. Boundary Values . . . . 661.4.2. Notation . . . . 671.4.3. Linear PDGL of 1st Order in 2 Variables . . . . 671.4.4. Linear PDGL of 2nd Order in n Variables: Classification . . . . 731.4.5. PDGL of 2nd Order in 2 Variables: Product Ansatz . . . . 78

2. Calculus of Variations . . . . 852.1. Brachistochrone Problem . . . . 852.2. Euler-Lagrange Differential Equations . . . . 872.3. Hamilton Function . . . . 91

3. Probability Theory . . . . 973.1. Combinatorics . . . . 973.2. Random Experiments and Probabilities . . . . 1003.3. Conditional Probability . . . . 1043.4. Independence . . . . 1073.5. Random Variables . . . . 1093.6. Distribution Function . . . . 1123.7. Expected Value and Variance . . . . 1163.8. Distributions . . . . 1233.8.1. Binomial Distribution . . . . 1233.8.2. Geometric Distribution . . . . 1253.8.3. Poisson Distribution . . . . 1263.8.4. Rectangular Distribution . . . . 1283.8.5. Exponential Distribution . . . . 1283.8.6. Normal Distribution . . . . 1283.8.7. Stability . . . . 1303.9. Limit Theorems . . . . 130

4. Statistics . . . . 1354.1. Measurement Series . . . . 1354.2. Empirical Distribution Function . . . . 1374.3. Test Distributions . . . . 1384.4. Estimation Procedures . . . . 1424.5. Maximum Likelihood Estimator . . . . 1444.6. Confidence Intervals . . . . 1464.7. Hypotheses and Parameter Testing . . . . 1524.7.1. (∗) Distribution Tests – χ2 Distribution Test . . . . 1594.8. Regression . . . . 1614.8.1. Correlation . . . . 1614.8.2. Gaussian Method of Least Squares . . . . 1624.9. Reduction to Linear Regression . . . . 165

A. Real Solutions of Linear Differential Equations . . . . 169B. Greek Alphabet . . . . 171C. Standard Normal Distribution . . . . 173D. Quantiles of the t-Distribution . . . . 175E. Quantiles of the χ2-Distribution . . . . 177

References . . . . 179Index . . . . 180

Differential Equations

• A differential equation is an equation that involves an unknown function and its derivatives.

• An example of a second-order differential equation is: u''(x) + 4 * u(x) = 0, to which a solution could be u(x) = cos(2x).

• Fundamental questions regarding differential equations include:

  • Existence: Does a solution exist?

  • Uniqueness: Are there multiple solutions?

  • Methods: How can we determine the solution(s)?

• The unknown function must be defined on an open interval in R or an open set in R^m.

• The order of a differential equation is determined by the highest order derivative present.

  • Example of order 2: u''(x) + 4 * u(x) = 0.

  • Example of order 1: u'(x) + (1/x) * u(x) = 0.

• A solution of an n-th order differential equation must be n-times differentiable.

• Boundary conditions can be provided for the unknown function and its derivatives at specific points, known as boundary values.

• Initial value problems: Specify conditions at the starting point (typically time). Example:

  • u''(t) + 4 * u(t) = 0 in (0,∞), u(0) = 1, u'(0) = 0.

• Boundary value problems: Specify conditions at two boundary points.

  • e.g., u''(t) + 4 * u(t) = 0 in (0,∞), u(0) = 1, u(π) = 2 has no solutions due to contradiction of periodicity.

Ordinary Differential Equations

Notation

• I denotes an open interval (e.g., I = (a, b)).

• For differentiable functions u: I → R, u' denotes the derivative of u.

• The notation for the n-th derivative is u^(n).

Fundamental Definitions

• An ordinary differential equation of n-th order can always be expressed in the form F(x, u(x), u'(x), ..., u^(n)(x)) = 0 in I by defining a mapping F: I × R^(n+1) → R.

• An equation is called an implicit differential equation if it takes this form.

• If it's solved for the highest derivative, it's an explicit differential equation.

Existence and Uniqueness of Solutions

• For first-order ordinary differential equation systems, important theorems include:

  • Peano's Existence Theorem: If the input function F is continuous, then a solution exists in a closed interval.

  • Picard-Lindelöf Theorem: Provides conditions under which the solution is unique and exists for every point in the interval if the Lipschitz condition is satisfied.