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Basic Concepts – In this chapter we introduce many of the basic concepts and definitions that are encountered in a typical differential equations course. We will also take a look at direction fields and how they can be used to determine some of the behavior of solutions to differential equations.

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See Section 1 on page , and equation [eq:euler] below. Table 1 and Figure 1 are referenced too.

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Mathematics

Inline math: the area of a circle is A = πr2, and x ∈ ℝ.

$$\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}.$$

eiπ + 1 = 0.

$$\begin{align} (a+b)^2 &= a^2 + 2ab + b^2 \label{eq:binom}\\ &= a^2 + b^2 + 2ab . \end{align}$$

$$f(x) = \begin{cases} x^2 & \text{if } x \ge 0,\\ -x & \text{otherwise.} \end{cases}$$

$$M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},\qquad \mathop{\mathrm{Tr}}M = 5, \qquad \left\lVert v \right\rVert = \sqrt{\textstyle\sum_i v_i^2}.$$

Theorems and proofs

Definition 1. A function f : ℝ → ℝ is continuous at a if limx → af(x) = f(a).

Theorem 1 (Pythagoras). In a right triangle, a2 + b2 = c2.

Lemma 2Every convergent sequence is bounded.

Proof. Standard; omitted here as a placeholder. ◻

Remark 1. Theorem environments convert unevenly across pandoc targets—worth checking.

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def hello(name):
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import numpy as np
def softmax(x):
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    return e / e.sum()

Miscellany


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An appendix section

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9 D. E. Knuth, The TeXbook, Addison–Wesley, 1984. L. Lamport, LaTeX: A Document Preparation System, 2nd ed., Addison–Wesley, 1994.


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