Equations and Inequalities

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Diffrentiation and integration

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On the Sunday morning, I'd like make new post about Integral and diffrentiation

Calculus – differentiation, integration etc. – is easier than you think. Here's a simple example: the bucket at right integrates the flow from the tap over time. The flow is the time derivative of the water in the bucket. The basic ideas are not more difficult than that. Calculus analyses things that change, and physics is much concerned with changes. For physics, you'll need at least some of the simplest and most important concepts from calculus. Fortunately, one can do a lot of introductory physics with just a few of the basic techniques.
So stick with us: differentiation really is just subtracting and dividing, and integration really is just multiplying and adding. This short introduction is no substitute, however, for a good high school calculus course: we shall take some short cuts of which mathematicians may disapprove.
Differentiation: How rapidly does something change? T…
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IMO 2014 problems and Solutions

On, Thursday, 12 May 2016. 1 : 48 pm. i'd like to make an solution of math which i copy from the book.

Problem 1. Let a0 < a1 < a2 . . . be an infinite sequence of positive integers. Prove that there exists a unique integer n ≥ 1 such that an < a0 + a1 + a2 + · · · + an n ≤ an+1.
 Problem 2. Let n ≥ 2 be an integer. Consider an n × n chessboard consisting of n 2 unit squares. A configuration of n rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer k such that, for each peaceful configuration of n rooks, there is a k × k square which does not contain a rook on any of its k 2 unit squares.
 Problem 3. Convex quadrilateral ABCD has ∠ABC = ∠CDA = 90◦ . Point H is the foot of the perpendicular from A to BD. Points S and T lie on sides AB and AD, respectively, such that H lies inside triangle SCT and ∠CHS − ∠CSB = 90◦ , ∠T HC − ∠DT C = 90◦ . Prove that line BD is tangent to the circumcircle of triangle T SH…
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Diffrential Equations

In this post, I'll introduce about Diffrential Equations. 
First-order differential equations Reference: Boyce and DiPrima, Chapter 2 The general first-order differential equation for the function 𝑦 = 𝑦(𝑥) is written as 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥, 𝑦), (2.1) where 𝑓(𝑥, 𝑦) can be any function of the independent variable 𝑥 and the dependent variable 𝑦. We first show how to determine a numerical solution of this equation, and then learn techniques for solving analytically some special forms of (2.1), namely, separable and linear first-order equations. 2.1 The Euler method view tutorial Although it is not always possible to find an analytical solution of (2.1) for 𝑦 = 𝑦(𝑥), it is always possible to determine a unique numerical solution given an initial value 𝑦(𝑥0) = 𝑦0, and provided 𝑓(𝑥, 𝑦) is a well-behaved function. The differential equation (2.1) gives us the slope 𝑓(𝑥0, 𝑦0) of the tangent line to the solution curve 𝑦 = 𝑦(𝑥) at the point (𝑥0, 𝑦0). With a small step…
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