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Ordinary Differential Equations

Ordinary Differential Equations
A differential equation is an equation with $\frac{d^n}{dx^n}$ terms. Let's start with a simple equation with two variables $x$ and $y$.

$y = ae^x + be^{3x}$

If we differentiate this in $x$, a first order differential equation comes as

$\frac{dy}{dx} = ae^x + 3be^{3x}$

If we differentiate it once more in terms of $x$, a second order differential equation is produced as

$\frac{d^2y}{dx^2} = ae^x + 9be^{3x}$

By solving there three equations, we can get an equation without those constant terms $a$ and $b$ and that is called an ordinary differential equation.

Big O Notation

What is Big O???
Big O notation is a technique used to describe the complexity of an algorithm. It is very useful when evaluating performance wise of different algorithms doing the same task. In this post, I'm not going deep into the programming side but let's take a code snippet as an example.

In this code, we are trying to add up all the natural numbers up to 100. There are two algorithms doing the same job but they perform differently.
#***********************## Method 1n=100sum=0foriinrange(0,n+1):sum=sum+int(i)printsum#***********************## Method 2printstr(n*(n+1)/2)#***********************#

Taylor Series

What is Taylor Series???
Taylor Series in simple terms, is a way of finding the value of a function at any point. To calculate and formulate the answer, all we need is the function value at a single point and the values of all its derivatives at that point. The general expression for Taylor Series is expressed as follows.
$$f(x) = \sum_{k = 0}^{\infty}\frac{1}{k!}(x - a)^kf^k(a)$$ Note that we need to know what the function values are at $f(a)$, $f'(a)$, $f''(a)$, ...

Functions

Image
Definition
Let A and B be two subsets of ℝ
A relation from A to B is a rule which assigns a member of B to each member of A.
Definition
Let A and B are two subsets of ℝ
A function from A to B is a rule which assigns a unique member of B of each number of A.

Intervals

Defenition
Let A ≤ R; A is said to be an interval if following condition is satisfied.

If a, b ∈ A with a < b, then
a < x < b ⇒ x ∈ A

Eg: A = {1, 2, 3, 4}
2, 3 ∈ A & 2 < 2.5 < 3 & 2.5 ∉ A

Therefore A is not an interval

Inequalities

Definition
ℝ+ = {a | a ∈ ℝ and a is located to the R.H.S. of 0 in the number line}
ℝ- = {a | a ∈ ℝ and a is located to the L.H.S. of 0 in the number line}
{0} = The single point 0
Then, ℝ = ℝ- ∪ {0} ∪ ℝ+

Irrational Numbers

In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable(and the rationals countable) it follows that almost all real numbers are irrational.
Note: ∃ a (There is...)∃ a1, a2, a3 ... (There are...)