Help on Differential Equation

Before we start with differential equation's let us first understand what is an equation?

What is an Equation??

A statement asserting the equality of two expressions, usually written as a linear array of symbols that are separated into left and right sides and joined by an equal sign is called an Equation.

Let us understand this with the help of the below example:

Example:
If one number is thrice the other and their sum is 60, find the numbers.
Solution:
Let the numbers be x and y.
x is 3 times y x = 3y …(1)
Sum of x and y is 60
x + y = 60 …(2)
Putting the value of x from (1) in (2), we get,
3y + y = 60
4y = 60
y = 15
Substituting y = 15 in (1), we get,
x = 3 x 15
= 45

The required numbers are 15 and 45.

What is Differential Equation:-

An equation involving problem on derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.

Below is one example on Differential Equation:

A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an Ordinary Differential Equation.

Equations involving derivatives with respect to more than one independent variables, called a Partial Differential Equation.

If you liked this please leave a comment. Will be back with some more topics that can make our life easier in understanding concepts in Math and solve problems.

Empirical probability

A form of probability that is based on some event occurring, which is calculated using collected empirical evidence. An empirical probability is closely related to the relative frequency in a given probability distribution.

In order for a theory to be proved or disproved, empirical evidence must be collected. An empirical study will be performed using actual market data. For example, many empirical studies have been conducted on the capital asset pricing model (CAPM), and the results are slightly mixed.

In some analyses, the model does hold in real world situations, but most studies have disproved the model for projecting returns. Although the model is not completely valid, that is not to say there is no utility associated with using the CAPM. For instance, the CAPM is often used to estimate a company's weighted average cost of capital.

An advantage of estimating probabilities using empirical probabilities is that this procedure is relatively free of assumptions. For example, consider estimating the probability among a population of men that they satisfy two conditions:

• That they are over 6 feet in height
• That they prefer strawberry jam to raspberry jam.

A direct estimate could be found by counting the number of men who satisfy both conditions to give the empirical probability the combined condition. An alternative estimate could be found by multiplying the proportion of men who are over 6 feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.

Conic Section

The locus of a point moving on a plane such that its distances from a fixed point and a fixed straight line on the plane are in constant ratio e, is called a conic.

The fixed point is called the focus and is usually denoted by S. The fixed straight line is called the directrix. The constant ratio e is called the eccentricity. The straight line on the plane passing through the focus and perpendicular to the directrix is called the axis. Therefore, the locus of a point moving on a plane such that SP/ P M = e , where PM is the perpendicular distance from p to the directrix, is called a conic.

• if e= 1 the conic is called a parabola
• if 0 <1>the conic is called an ellipse
• if e> 1 the conic is called a hyperbola.

Line and Line Segment

A line is a set of points on a directly path that extends to infinity. Any two nodes on the line can be used to name it. This type of line is known as line DE. A line segment is a division of a line that has two end points. Here we will see this line segment and line . A ray is also a section of a line.

Line, Line Segment

Line segments having two end points. These endpoints are used to put the name of the line segments. This line segment is known as segment AB.

Line Segment Angle

Definition

If V is a vector space over R or C, and L is a subset of V then L is a line segment if L can be parameterize as,

For some vectors u, v ε V, in which case the vectors u and u+v called the end points of L.

Occasionally one necessitates difference between open and the closed line segments. Then one explained the closed line segments, and the open line segment of a subset L.

More frequently than above, the concept of a line segment can define in an arranged geometry. (

Properties of Line Segments

• A line segments are connected to the non-empty set.
• If V is a topological vector space, then a protected line segment is a closed set in V.

Discrete Probability Distribution

Discrete Probability Distribution

A discrete random variable assumes each of its values with a certain probability,

Let X be a discrete random variable which takes values x₁, x₂, x₃,…x_n where p_i = P{X = x_i}

Then

X : x₁ x₂ x₃ .. x_n

P(X): p₁ p₂ p_{3 .....} p_n

is called the probability distribution of x.

Note 1:

In the probability distribution of x

Note 2:

P{X = x} is called probability mass function.

Note 3:

Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can have a formula in the form of a function represented by f(x) usually called the probability density function.

Mixture problems

The Mixture problems are part of the algebra word problems. The mixture problems deals with two or more things mixed in different ratio and ask to find price, percentage and some other quantities. For solving mixture problems we need to know the following things,
The total amount = amount of first mixture + amount of second mixture .This formula used to find the unknown mixture amount.
We can form the equation to find unknown value by using the following formula,
(Cheaper value / dearer value) = (mean value –cheaper value)/ (dearer value –mean value)
Let us understand this with the following example on mixture problem.
f Real gold is 92.5% pure gold. How many pounds of Real Gold needs to be mixed to a 90% Gold alloy to obtain a 500g of 91% gold alloy?
Solution:
Let us take x and y be the weights of real gold and of the 90% alloy to make the 500 pounds at 91%.
That is x + y =500
The formula is,
amount₁ (%) + amount₂ (%) = (amount₁ + amount₂) (%)

92.5 % of x + 90% of y = 91% of 500
Substitute y = (500 – x) in the above equation,
We get,
92.5% of x + 90% of (500 - x) = 91% of 500
(92.5x)/100 + (90(500-x))/100 = (91(500))/100
Multiply by 100 on both sides,
92.5x+ 45000-90x=45500
Subtract 45000 on both sides,
92.5x-90x=45500-45000
2.5x=500
Divide by 2.5 on both sides,
X = 500/2.5
X = 200
Answer : 200 pounds of gold needs to make 91% of gold alloy.

Notations

A complex number is a number combination of a real and imaginary part. It can be written in the form of a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property of i ² = −1. Complex numbers are containing the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

Notation:

The set of all complex numbers is usually denoted with an imaginary and real part.
Complex numbers are usually written in the form of a and b.
The form is a + bi
From this part a and b are real numbers, and i is the imaginary part, which has the property i 2 = −1. The real number a is called as real part of the complex number, and the real number b is called as imaginary part.

For example:

4 + 3i is a complex number, with real part 4 and imaginary part 3. If z = a + bi, the real part a is denoted Re(z), and the imaginary part b is denoted as Im(z).
In some disciplines, the imaginary part i is instead written as j, so complex numbers are sometimes contain as a + bj or a + jb.

History of Mathematical Notation:

Below list shows history of mathematical symbols for list of symbols in history of mathematical notation:

• Beginning of notation
• Greek notation
• Chinese notation
• Indian notation
• Beginning of Arabic numerals
• North African notation
• Pre-calculus
• Calculus
• Euler

Parameters in Mathematics

The parameter in mathematics is an amount that serves to transmit functions and variables with a general variable. When a connection would be hard to explain with an equation. In different context the word may contain particular use. A variable parameter is a proper parameter whose name is close to the similar memory position as that named by the real parameter at the moment the call is complete.

Parameters in Mathematics

Mathematical functions contain one or more arguments that are chosen in the meaning by variables, while their meaning can also have parameters. The variables are mention in the record of point of view that the meaning takes, however the parameters are not For example single can define a common quadratic function through defining

f(x) = ax² + bx + c;

Here the variable x designate the function argument, however a, b, and c are parameters that establish which quadratic function single is considering. The parameter might be included into the function name to designate its dependence on the parameter; for case single may describe the bottom a logarithm by

Where a is a parameter that indicate which logarithmic function is individual used; it is not an row of the function, and will for case be a constant while considering the derived

Mathematical analysis

The integrals dependent on a limit are often measured. These are of the form

In this method, t is the row of the function F and on the right-hand parts the parameter on which the basic depends. When evaluate the integral, t is detained constant, and therefore it considered a parameter. If we are involved in the value of F for special values of t, then, we now regard as it to be a variable. The amount x is a dummy variable or variable of integration.

Triangle and its Relations

A triangle is a three sided geometric polygon with three sides and angles. In some of the triangles of every one of the faces will be equal area which is known as the equilateral triangle. In right angle triangle the longest side is named as the hypotenuse and the other two sides be opposite side and adjacent side. All the triangles are biocentrism and convex. This area is known as the interior area of the triangle and the remaining portion is known as the exterior. The center of triangle can be found by intersection of three internal angle bisectors.

The triangle has following relations:

1. Circumcenter:
In the triangle relations circumcenter defined as a center of the circle where the three perpendicular bisectors intersect with each other.

2. Incenter:

Incenter is the point where the single bisectors intersect. In these triangle relations are the center of the circle that fits exactly inside the triangle.

3. Altitude:

Other one triangle relations altitudes is the "height" a segment that starts at a vertex and is perpendicular to the opposite side.

4. Median:

The triangle relations, such as median is known as the measure of distance between the midpoint of one side and the vertex of the opposite angle.

5. Angle bisector:

Angle bisector of triangle relations are nothing but the perpendicular bisector which cuts the given angle in to two equal half. The angle formed by this will equal.

6. Perpendicular bisector:

It cuts a segment in to equal half at a 90 degree angle, when the circumcenter is on the hypotenuse the circumscribe is outside of the triangle.

7. Pythagoras theorem:

Another important triangle relations of Pythagoras theorem is defined by that the square of the hypotenuse of the right angle triangle is equal to the sum of the square of the sides of the triangle. It can be applicable only in right angled triangles. By using this we can be able find the missing values. Suppose we know the value of two side of the triangle then we can be able to find the value of third side by using the Pythagoras theorem.

Tessellations

Patterns which fill a plane are called tessellations. A regular tessellation is a tessellation made up of regular polygons of the same size.

Examples of Geometry Tessellations:

Let us see geometry tessellation examples here.
Tessellations of equilateral triangles:

The above figure represents the example of tessellation of equilateral triangle.
Tessellations of squares:

The above figure represents the example of tessellation of Square.
Tessellations of Polygons:

The above figure represents the example of tessellation of polygon.
Looking at the above examples, We notice that the squares are lined up with each other as in the figure. Here there is no gap between the tiles while filling the plane with the square shaped tiles.

Deductive Reasoning

A deductive argument is an argument in which it is consideration that the premises provide a guarantee of the truth of ending. In the argument, the conclusion is depending on the premises. The conclusion of deductive reasoning will be valid when the premises are true. In deductive reasoning we can say that the arguments can be valid or invalid but we can never say that the arguments are true or false.

Rules for deductive reasoning:

• Every deduction should contain three and only three distinct terms.

• In an argument, the middle term must be distributed at least once in the premises

• If one premise in an argument is negative, the conclusion, if any must be negative

• If one premises is particular, the conclusion, if any, should be particular

• If both the premises in the statements are negative, then no conclusion can be drawn

• If both the premises are particular, no conclusion can be derived

• The term which is not distributed in the premises, cannot be distributed in the conclusion.

Example: An example of the deductive argument with a deductive reasoning, Statement:

• All thoughts are actions

• Some actions are not disasters

The conclusion is derived by checking whether it is logically follows the given two statements.

Ratio

Ratio means pairs of numbers and they are used to formulate comparisons of numbers. A ratio can be used to compares two numbers using a fraction.

Listed below are some examples on how to find a Ratio.

Example 1) A special mixture contains dal, grams and wheat in the ratio of 2:3:6. If a bag contains 6 pounds of dal, how much wheat it contains?

Solution:

Let x = amount of wheat

write down the items in the ratio as a fraction.

Dal/wheat = 2/6 = 6/x

Solve the equation by cross multiplying it

2*x = 6*6
2x = 12
x = 12/2
Therefore x = 6

Hence the bag contains 6 pounds of wheat

Multuplications of Fractions

Fraction is an equal part of one whole object.

Fraction can be represented as "p/q" where 'p' denotes the value called numerator and 'q' denotes the value called denominator and q not equal to zero.

There are 3 simple steps to Multiply Fractions:

• Multiply the top numbers ( the numerators)
• Multiply the bottom numbers ( the denominators)
• Simplify the fraction

Examples on multiplication of fractions:

Listed below are some examples on multiplication of fractions.

Example 1) Multiply 6/10 and 16/12
6/10*16/12 = (6*16)/(10*12)
= 96/120
= 4/5
Hence, the product of 6/10 and 16/12 is 4/5

Example 2) Multiply 4/10 and 6/8
4/10*6/8 = (4*6)/(10*8)
= 24/80
= 3/10
Hence, the product of 4/10 and 6/8 is 3/10