Coefficient of Variation Formula
Last Updated :
23 Jul, 2024
Coefficient of deviation in statistics is explained as the ratio of the standard deviation to the arithmetic mean, for instance, the expression standard deviation is 15 % of the arithmetic mean is the coefficient variation.
In this article, we have covered the Coefficient of Variation Definition, its related formulas, examples and others in detail.
What is Coefficient of Variation?
The coefficient of deviation is the dimension of comparable variability. The coefficient of deviation is the percentage of the expected deviation from the standard.
It's very useful if one wants to compare the results from two different research or tests that consist of two different results. For example, if comparing the results of two different matches that have two completely different scoring methods, like if model X has a CV of 15% and model Y has a CV of 30%, it would be conveyed that model Y has more deviation, comparable to its mean. It enables us to supply relatively simple and quick tools that help us to compare the data of different series.
Formula to calculate the coefficient of variation is:
Coefficient of Variation = (Standard Deviation / Mean) × 100
In symbols: CV = (SD/x̄) × 100
Coefficient of Variation (In Finance)
It helps in the investment selection process that's why it is important in terms of finance. In the financial matrix, it shows the risk-to-reward ratio means here the standard deviation/ volatility show as the risk of the investment, and the mean is shown as the expected reward of the investment. The investors in the company identify the risk-to-reward ratio of each one of the security to develop an investment decision. In this, the low coefficient is not favourable when the average expected return is below the value of zero
The Formula for the coefficient of variation in the context of finance,
Coefficient of Variation = σ/μ × 100%
where,
- σ is Standard Deviation
- μ is Mean
Example for Coefficient of Variation Finance
An investor Sudhir wants to find new investments for his portfolio purpose. Where he was looking for secure investments that provides him a stable return. So, consider some of the following options for the investment.
- Stocks: Sudhir has gotten an offer for the stocks of XYZ.Pvt corporates is a very mature company with strong financial and operational performance. The volatility of this stock is 9% and the expected return in of 13%.
- Bonds: Sudhir is getting bonds with excellent credit ratings, which offered him an expected return of 5% with 3% of volatility.
- ETFs: Another option sudhir gets is an EXCHANGE-TRADED FUND which helps in the tracking of the S&P 500 index. An ETF offers him an expected return of 15% with a volatility of 8%.
Standard Deviation
Standard deviation formula helps us to find the values of a particular data that is dispersed. Simply, it is defined as the deviation of the data from an average mean. In this, higher values mean that the values are far from the average mean as well as the lower values mean that values are very close to their average mean. It is said that the value of standard deviation can never be negative.
Standard deviation is of two types
Population Standard Deviation
σ = √∑(X − μ)²/n
Sample Standard Deviation
s = √∑(X − x̄)²/n − 1
Notation for standard deviation,
- σ is Standard Deviation
- xi is Terms Given in the Data
- x̄ is Mean
- n is Total number of Terms
Steps for the calculation of the coefficient of variation are as follow
- Step 1: Firstly, get the standard deviation of the data provided.
- Step 2: After getting the standard deviation of the data we have to get the mean of the given data as we know that the coefficient of variation is the ratio between the standard deviation and the mean of the given data.
- Step 3: Now put both the standard deviation and the mean in the formula and multiply the coefficient by 100 is an optional step to get a percentage, as opposed to a decimal.
Example 1: The standard deviation and mean of the data are 8.5 and 14.5 respectively. Find the coefficient of variation.
Solution:
SD/σ = 8.5
mean/μ = 14.5
Coefficient of variation = σ/μ × 100%
= 5.5/14.5 × 100
Coefficient of variation = 58.6%
Example 2: The standard deviation and coefficient of variation of data are 1.4 and 26.5 respectively. Find the value of the mean.
Solution:
C.V = 26.5
SD/σ = 1.4
Mean/x̄ = ?
C.V = σ/x̄ × 100
26.5 = 1.4 / x̄ × 100
x̄ = 1.4/26.5 × 100
x̄ = 5.28
Example 3: If the mean and coefficient of deviation of data are 13 and 38 respectively, then locate the value of expected variation?
Solution:
C.V = 38
SD/σ = ?
Mean/x̄ = 13
C.V = σ/x̄ × 100
38 = σ/13 × 100
σ = 13 × 38/100
σ = 4.9
Example 4: The mean and standard variation of marks received by 40 students of a class in three subjects Mathematics, English and economics are given below.
| Subject | Mean | Standard deviation |
|---|
| Maths | 65 | 10 |
| English | 60 | 12 |
| Economics | 57 | 14 |
Which of the three subjects indicates the most elevated deviation and which indicates the most subordinate variation in marks?
Solution:
Coefficient of variation for maths = σ/x̄ × 100
σ = 10.
x̄ = 65
C.V = 10/65 × 100
Coefficient of variation for maths = 15.38%
Coefficient of variation for english = σ/x̄ × 100
σ = 12
x̄ = 60
C.V = 12/60 × 100
Coefficient of variation for english = 20%
Coefficient of variation for economics = σ/x̄ × 100
σ = 14
x̄ = 57
C.V = 14/57 × 100
Coefficient of variation for economics = 24.56%
The highest variation is economics.
And the lowest variation in maths.
Example 5: The following table gives the values of mean and friction of heights and weights of the 10th common students of a school.
| | Height | Weight |
|---|
| Mean | 157 cm | 56.50 kg |
| Variance | 72.25 cm | 28.09 kg |
Which is more varying than the other?
Solution:
Coefficient of variation for heights
Mean x̄1 = 157cm, variance σ1² = 72. 25 cm²
Therefore standard deviation σ1 = 8. 5
Coefficient of variation
C.V1 = σ/x̄ × 100
= 8.5/157 × 100
C.V1 = 5.41% (For heights)
Coefficient of variation for weights
Mean x̄2 = 56.50kg, variance σ2² = 28.09 kg²
Therefore standard deviation σ2 = 5.3kg
Coefficient of variation
C.V1 = σ/x̄ × 100
= 5.3/56.50 × 100
C.V2 = 9.38% (For weight)
C.V1 = 5.41% and C.V2 = 9.38%
Since C .V2 > C.V1, the weight of the students is more varying than the height.
Example 6: During a survey, 6 students were asked how many hours per day they study on average? Their answers were as follows: 3, 7, 5, 4, 3, 5. Evaluate the standard deviation.
Solution:
Find the mean of the data:
(3 + 7 + 5 + 4 + 3 + 5)/6
= 4.5
| X1 | X1 - x̄ | (X1 - x̄)² |
| 3 | -1.5 | 2.25 |
| 7 | 2.5 | 6.25 |
| 5 | 0.5 | 0.25 |
| 4 | -0.5 | 0.25 |
| 3 | -1.5 | 2.25 |
| 5 | 0.5 | 0.25 |
= 11.5
Sample standard deviation formula:
s = √∑(X − x̄)²/n − 1
= √(11.5/[6 - 1])
= √[2.3]
= 1.516
Example 7: Four friends were comparing their scores on a recent essay. Calculate the standard deviation of their scores: 7, 3, 4, 2.
Solution:
Find the mean of the data
(7 + 3 + 4 + 2)/4
= 4
| X1 | X1 - x̄ | (X1 - x̄)² |
| 7 | 3 | 9 |
| 3 | -1 | 1 |
| 4 | 0 | 0 |
| 2 | -2 | 4 |
= 14
Population standard deviation
σ = √∑(X − x̄)²/n
= √(14/[4])
= √[10]
= 3.162
Worksheet: Coefficient of Variation
Problem 1: Given the dataset: {10,12,14,16,18}. Calculate the Coefficient of Variation.
Problem 2: Given the dataset: {50,60,70,80,90}. Calculate the Coefficient of Variation.
Problem 3: Daily temperatures in degrees Celsius: {20,22,24,26,28}. Calculate the Coefficient of Variation.
Problem 4: Blood pressure readings: {120,125,130,135,140}. Calculate the Coefficient of Variation.
Problem 5: Given the dataset: {1,2,4,8,16,32}. Calculate the Coefficient of Variation.
Problem 6: Dataset 1: {6,8,10,12,14}, Dataset 2: {12,16,20,24,28}. Calculate the CV for both datasets and determine which has higher relative variability.
Problem 7: Dataset 1: {20,22,24,26,28}, Dataset 2: {30,33,36,39,42}. Calculate the CV for both datasets and determine which has higher relative variability.
Problem 8: Monthly sales figures in dollars: {1000,1500,2000,2500,3000}. Calculate the Coefficient of Variation.
Problem 9: Weights of packages in kg: {2,2.5,3,3.5,4}. Calculate the Coefficient of Variation.
Problem 10: Given the dataset: {4,8,12,16,20,24}. Calculate the Coefficient of Variation.
Summary
Coefficient of Variation (CV) is a standardized measure of the dispersion of a probability distribution or frequency distribution. It is often used to compare the degree of variation between different datasets, especially when the units of measurement differ or the means of the datasets are significantly different.
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Algebra Formulas - List of all Algebra FormulasAlgebra formulas are mathematical expressions that help solve problems involving variables and constants. They often represent relationships between quantities and can be used to simplify calculations. This article provides a comprehensive overview of all algebra formulas taught from Class 9 through
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Polynomial FormulaThe polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. The General Formula of a Polynomial:f(x) = an​xn + an−1​xn−1 + ⋯ + a1​x + a0​Where,an​, an−1​, …, a1​, a0​ a
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Factorization of PolynomialFactorization in mathematics refers to the process of expressing a number or an algebraic expression as a product of simpler factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and we can express 12 as 12 = 1 × 12, 2 × 6, or 4 × 3.Similarly, factorization of polynomials involves expre
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What is Factoring Trinomials Formula?A Trinomial is a polynomial with three terms. Examples of Trinomial are x+y+z, x2+2x+2, x+y-1 etc. A Trinomial can be of two types. They are Perfect Square Trinomial and Non-Perfect Square Trinomial. Factoring a polynomial is nothing but writing the expression polynomial as a product of two or more
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a2 - b2 Formulaa2 - b2 formula in Algebra is the basic formula in mathematics used to solve various algebraic problems. a2 - b2 formula is also called the difference of squares formula, as this formula helps us to find the difference between two squares without actually calculating the squares. The image added bel
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Difference of CubesDifference of Cubes is the formula in mathematics that is used to simplify the difference between two cubes. This formula is used to solve the difference of cubes without actually finding the cubes. This formula factorizes the difference of a cube and changes it into other forms. The difference of c
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Discriminant Formula in Quadratic EquationsAlgebra can be defined as the branch of mathematics which deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for t
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Sum of Arithmetic Sequence FormulaAn arithmetic sequence is a number series in which each subsequent term is the sum of its preceding term and a constant integer. This constant number is referred to as the common difference. As a result, the differences between every two successive terms in an arithmetic series are the same.If the f
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Function Notation FormulaA function is a type of operator that takes an input variable and provides a result. When one quantity is dependent on another, a function is created. An interesting property of functions is that each input corresponds to a single output. In other words, such an operator between two sets, say set A
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Binomial Distribution in ProbabilityBinomial Distribution is a probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is useful for calculating the probability of a specific number of successes in sce
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Binomial Expansion FormulasBinomial expansion formula is a formula that is used to solve binomial expressions. A binomial is an algebraic expression with two terms. For example, x + y, x - a, etc are binomials. In this article, we have covered the Binomial Expansion definition, formulas, and others in detail.Table of ContentB
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Binomial TheoremBinomial theorem is a fundamental principle in algebra that describes the algebraic expansion of powers of a binomial. According to this theorem, the expression (a + b)n where a and b are any numbers and n is a non-negative integer. It can be expanded into the sum of terms involving powers of a and
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FOIL MethodFOIL formula is used to perform multiplication between two binomials. A binomial is a polynomial with only two terms. Example: x+3, x2+4, 5x2+3x. A binomial is a 2 term algebraic expression that includes constants, variables, exponents and coefficients. Each letter in the FOIL represent steps to mul
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Exponential Decay FormulaExponential Decay Formula: A quantity is said to be in exponential decay if it decreases at a rate proportional to its current value. In exponential decay, a quantity drops slowly at first before rapidly decreasing. The exponential decay formula is used to calculate population decay (depreciation),
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Factorial FormulaThe factorial is one of the most fundamental mathematical operations in combinatorics, algebra, and number theory. Represented by an exclamation mark (!), the factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. It plays a crucial rol
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Combinations Formula with ExamplesCombinations are way of selecting items from a collection of items. Different groups that can be formed by choosing r things from a given set of n different things, ignoring their order of arrangement, are called combinations of n things taken r at a time.The number of all such combinations is calcu
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Fourier Series FormulaFourier Series is a sum of sine and cosine waves that represents a periodic function. Each wave in the sum, or harmonic, has a frequency that is an integral multiple of the periodic function’s fundamental frequency. Even though a Fourier series can include infinitely many harmonics, using just a few
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Maclaurin seriesPrerequisite - Taylor theorem and Taylor series We know that formula for expansion of Taylor series is written as: f(x)=f(a)+\sum_{n=1}^{\infty}\frac{f^n(a)}{n!}(x-a)^n Now if we put a=0 in this formula we will get the formula for expansion of Maclaurin series. T hus Maclaurin series expansion can b
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Coordinate Geometry
Trigonometry
30-60-90 FormulaA 30-60-90 triangle is a special type of right triangle with one angle measuring 30°, another 60°, and the third angle (the right angle) measuring 90°. The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. Here, a right triangle means
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Cofunction FormulasA trigonometric cofunction is defined as expressing a trigonometric angle ratio in terms of the other. It illustrates how sine, cosine, tangent, cotangent, secant, and cosecant relate to each other. The cofunction of an angle's complement is equal to that angle's trigonometric function. For example,
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What is Cos Square theta Formula?The equations that relate the different trigonometric functions for any variable are known as trigonometric identities. These trigonometric identities help us to relate various trigonometric formulas and relationships with different angles. They are sine, cosine, tangent, cotangent, sec, and cosec.
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What are Cosine Formulas?Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship betwe
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Cosecant FormulaCosecant is one of the six basic trigonometric ratios and its formula is cosecant(θ) = hypotenuse/opposite, it is also represented as, csc(θ). It is the inverse(reciprocal) ratio of the sine function and is the ratio of the Hypotenus and Opposite sides in a right-angle triangle. In this article, we
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Cotangent FormulaTrigonometry is an important branch of mathematics that deals with the relation between the lengths of sides and angles of a right-angled triangle. Sine, Cosine, tangent, cosecant, secant, and cotangent are the six trigonometric ratios or functions. Where a trigonometric ratio is depicted as the rat
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Tangent FormulasTangent Function is among the six basic trigonometric functions and is calculated by taking the ratio of the perpendicular side and the hypotenuse side of the right-angle triangle.In this article, we will learn about Trigonometric ratios, Tangent formulas, related examples, and others in detail.Tabl
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Cot Half Angle FormulaTrigonometry is a branch of mathematics that uses trigonometric ratios to determine the angles and incomplete sides of a triangle. The trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant are used to investigate this branch of mathematics. It's the study of how the sid
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2cosA cosB FormulaThe identity 2 cos A cos B = cos(A + B) + cos(A – B) is one of the important product-to-sum formulas in trigonometry. This identity is used to convert a product of cosine functions into a sum of cosines, which can simplify the process of solving trigonometric equations, evaluating integrals, and per
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Multiple Angle FormulasTrigonometry is one of the important topics in mathematics that is used in various fields. The trigonometric formulae are applied and used in various formulae, derivations, etc. This article is about the multiple angle formulae in trigonometry where we find sine, cosine, and tangent for multiple ang
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Double Angle Formula for CosineDouble angle formula for cosine is a trigonometric identity that expresses cosâ¡(2θ) in terms of cosâ¡(θ) and sinâ¡(θ) the double angle formula for cosine is, cos 2θ = cos2θ - sin2θ. The formula is particularly useful in simplifying trigonometric expressions and solving equations involving trigonometri
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Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
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Complex Number
Calculus
Limit FormulaLimits help us comprehend how functions behave as their inputs approach certain values. Think of a limit as the destination that a function aims to reach as the input gets closer and closer to a specific point.In this article, we will explore the essential limit formulas that form the backbone of ca
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Average and Instantaneous Rate of ChangeThe average rate of change represents the total change in one variable in relation to the total change of another variable. Instantaneous rate of change, or derivative, measures the specific rate of change of one variable in relation to a specific, infinitesimally small change in the other variable.
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Calculus | Differential and Integral CalculusCalculus was founded by Newton and Leibniz. Calculus is a branch of mathematics that helps us study change. It is used to understand how things change over time or how quantities grow, shrink, or accumulate. There are two main parts of calculus:Differential Calculus: It helps us calculate the rate o
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Total DerivativeTotal Derivative of a function measures how that function changes as all of its input variables change. For function f at a point is an approximation near the point of the function w.r.t. (with respect to) its arguments (variables).It is an approximation of the actual change in the function and is u
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Difference Quotient FormulaThe Difference Quotient Formula is a part of the definition of a function derivative. One can get derivative of a function by applying Limit h tends to zero i.e., h ⇢ 0 on difference quotient function. The difference quotient formula gives the slope of the secant line. A secant line is a line that p
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Chain Rule: Theorem, Formula and Solved ExamplesThe Chain Rule is a way to find the derivative of composite functions. It is one of the basic rules used in mathematics for solving differential equations. It helps us to find the derivative of composite functions such as (3x2 + 1)4, (sin 4x), e3x, (ln x)2, and others. Only the derivatives of compos
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Implicit DifferentiationImplicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
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Antiderivative: Integration as Inverse Process of DifferentiationAn antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that:d/dx[F(x)] = f(x)In other words, F(x) is a function whose derivative is f(x).Antiderivatives include a family of functions t
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Integration FormulasIntegration Formulas are the basic formulas used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are beneficial for finding
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Integration by PartsIntegration by Parts or Partial Integration, is a technique used in calculus to evaluate the integral of a product of two functions. The formula for partial integration is given by:∫ u dv = uv - ∫ v duWhere u and v are differentiable functions of x. This formula allows us to simplify the integral of
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Integration by Substitution FormulaThe process of finding the anti-derivative of a function is the inverse process of differentiation i.e. finding integral is the inverse process of differentiation. Integration can be used to find the area or volume of a function with or without certain limits or boundaries It is shown as∫g(x)dx = G(
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Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
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Area Under CurveArea Under Curve is area enclosed by curve and the coordinate axes, it is calculated by taking very small rectangles and then taking their sum if we take infinitely small rectangles then their sum is calculated by taking the limit of the function so formed.For a given function f(x) defined in the in
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Differentiation and Integration FormulaDifferentiation and Integration are two mathematical operations used to find change in a function or a quantity with respect to another quantity instantaneously and over a period, respectively. Differentiation is an instantaneous rate of change, and it breaks down the function for that instant with
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Differential EquationsA differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur
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