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An equation says that two things are equal. It will have an equals sign "=" like this: = 10 − 1 That equation says: what is on the left (7 + 2) is equal to what is on the right (10 − 1) V.RAMYA 3

There are two types of equations: Algebraic Equations Transcendental Equations ALGEBRAIC EQUATIONS : A finite combination of numbers, variables, an operating symbols and an equal sign. EX : 2X+5 = 15 There are two types of equations: Algebraic Equations Transcendental Equations ALGEBRAIC EQUATIONS : A finite combination of numbers, variables, an operating symbols and an equal sign. EX : 2X+5 = 15 V.RAMYA 4

* LINEAR EQUATION *QUARDRATIC EQUAITON *CUBIC EQUATION * QUARTIC EQUATION *POLYNOMIAL EQUATION *RATIONAL EQUATION *IRRATIONAL EQUATION * LINEAR EQUATION *QUARDRATIC EQUAITON *CUBIC EQUATION * QUARTIC EQUATION *POLYNOMIAL EQUATION *RATIONAL EQUATION *IRRATIONAL EQUATION V.RAMYA 5

Linear Equation: Linear equations are equations of the type ax + b = 0 with a ≠ 0. EX: * 5x + 1 = 6 * 2x + 3 = 0 Linear Equation: Linear equations are equations of the type ax + b = 0 with a ≠ 0. EX: * 5x + 1 = 6 * 2x + 3 = 0 V.RAMYA 6

a) The equation with one variable: An equation who have only one variable. Examples: 8a - 8 = 0 9a = 72. b) The equation with two variables: An equation who have only two types of variable in the equation. Examples: 8a - 8d = 74, 9a+6b-82=0. a) The equation with one variable: An equation who have only one variable. Examples: 8a - 8 = 0 9a = 72. b) The equation with two variables: An equation who have only two types of variable in the equation. Examples: 8a - 8d = 74, 9a+6b-82=0. V.RAMYA 7

c) The equation with three variables: An equation who have only three types of variable in the equation. Examples: 5x + 7y - 6z = 12 13a - 8b + 31c = 74 6p + 14q -7r + 82 = 0. c) The equation with three variables: An equation who have only three types of variable in the equation. Examples: 5x + 7y - 6z = 12 13a - 8b + 31c = 74 6p + 14q -7r + 82 = 0. V.RAMYA 8

Quadratic Equations: Quadratic equations are equations of the type ax 2 + bx + c = 0, with a ≠ 0. EX: 4x 2 +2X-2=0 Incomplete quadratic equations : ax 2 = 0 ax 2 + bx = 0 ax 2 + c = 0 Quadratic Equations: Quadratic equations are equations of the type ax 2 + bx + c = 0, with a ≠ 0. EX: 4x 2 +2X-2=0 Incomplete quadratic equations : ax 2 = 0 ax 2 + bx = 0 ax 2 + c = 0 V.RAMYA 9

Cubic Equations : Cubic equations are equations of the type ax 3 + bx 2 + cx + d = 0, with a ≠ 0. EX: 5x 3 - 3x 2 + 2x + 6 = 0 Cubic Equations : Cubic equations are equations of the type ax 3 + bx 2 + cx + d = 0, with a ≠ 0. EX: 5x 3 - 3x 2 + 2x + 6 = 0 V.RAMYA 10

Quartic Equations : Quartic equations are equations of the type ax 4 + bx 3 + cx 2 + dx + e = 0, with a ≠ 0. Biquadratic Equations: Biquadratic equations are quartic equations that do not have terms with an odd degree. ax 4 + bx 2 + c = 0, with a ≠ 0. Quartic Equations : Quartic equations are equations of the type ax 4 + bx 3 + cx 2 + dx + e = 0, with a ≠ 0. Biquadratic Equations: Biquadratic equations are quartic equations that do not have terms with an odd degree. ax 4 + bx 2 + c = 0, with a ≠ 0. V.RAMYA 11

A rational equation is an equation with rational expressions on either side of the equals sign.rational expressions An equation that has a variable in a denominator Example: A rational equation is an equation with rational expressions on either side of the equals sign.rational expressions An equation that has a variable in a denominator Example: V.RAMYA 12 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

Irrational equation is equation, that contains variable under radical or variable is a base of power with fractional exponent. Example: Irrational equation is equation, that contains variable under radical or variable is a base of power with fractional exponent. Example: V.RAMYA 13 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

POLYNOMIAL EQUATIONS Polynomial equations are in the form P(x) = 0, where P(x) is a polynomial. POLYNOMIAL EQUATIONS Polynomial equations are in the form P(x) = 0, where P(x) is a polynomial. V.RAMYA 14 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

The first one isn’t a polynomial because it has a negative exponent and all exponents in a polynomial must be positive. The second one isn’t a polynomial because polynomial equation does not contain roots. The third one isn’t a polynomial. variables in the denominator of a fraction The first one isn’t a polynomial because it has a negative exponent and all exponents in a polynomial must be positive. The second one isn’t a polynomial because polynomial equation does not contain roots. The third one isn’t a polynomial. variables in the denominator of a fraction V.RAMYA 15 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 16 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

The transcendental equations are equations that include transcendental functions like Exponential Logarithmic Trigonometric The transcendental equations are equations that include transcendental functions like Exponential Logarithmic Trigonometric V.RAMYA 17

Exponential Equations Exponential equations are equations in which the unknown appears in the exponent. Exponential Equations Exponential equations are equations in which the unknown appears in the exponent. V.RAMYA 18

Logarithmic Equations Logarithmic equations are equations in which the unknown is affected by a logarithm. Logarithmic Equations Logarithmic equations are equations in which the unknown is affected by a logarithm. V.RAMYA 19

TRIGONOMETRIC EQUATIONS : Trigonometric equations are the equations in which the unknown is affected by a trigonometric function. Example: TRIGONOMETRIC EQUATIONS : Trigonometric equations are the equations in which the unknown is affected by a trigonometric function. Example: V.RAMYA 20

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DegreeNameExample 0Constant7 1Linearx+3 2Quadraticx 2 −x+2 3Cubicx 3 −x Quartic6x 4 −x 3 +x−2 5Quinticx 5 −3x 3 +x 2 +8 V.RAMYA 22

DegreeNameExample 6 sexticsextic (or) hexichexic x 6 −3x 3 +x septic septic (or)hepticheptic x 7 −3x 5 +x octic x 8 −3x 7 +x nonic x 9 −3x 8 +x decic x 10 −3x 3 +x 2 +9 V.RAMYA 23

ExpressionDegree log(x)0 exex ∞ 1/x−1 1/2 V.RAMYA 24

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An equation involving derivatives of one or more dependent variables with respect to one or more independent variables is called a differential equation. An equation involving derivatives of one or more dependent variables with respect to one or more independent variables is called a differential equation. V.RAMYA 26

Types of differential equation ordinary differential equation partial differential equation Order, degree linearity Types of differential equation ordinary differential equation partial differential equation Order, degree linearity V.RAMYA 27

If a differential equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable, then it is called an ordinary differential equation (ODE) V.RAMYA 28 Y is dependent variable X is an independent variable

If a differential equation contains partial derivatives of one or more dependent variables with respect to two or more independent variables, then it is called a partial differential equation (PDE). u is a dependent x 1 and x 2 Variable is an independent variable If a differential equation contains partial derivatives of one or more dependent variables with respect to two or more independent variables, then it is called a partial differential equation (PDE). u is a dependent x 1 and x 2 Variable is an independent variable V.RAMYA 29

Ordinary differential equation Partial differential equation Y is dependent variable X is an independent variable u is a dependent Variable x 1 and x 2 is an independent variable d ∂ ramya 30

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The order of a DE is determined by the highest derivative in the equation. V.RAMYA 32

Degree of Differential Equation: The degree of differential equation is represented by the power of the highest order derivative in the given differential equation. Degree of Differential Equation: The degree of differential equation is represented by the power of the highest order derivative in the given differential equation. V.RAMYA 33

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1.If the dependent variable (y) and its derivatives are of the first degree, and each coefficient depends only on the independent variable (x), then the differential equation is linear. 2.Otherwise, the differential equation is nonlinear. 1.If the dependent variable (y) and its derivatives are of the first degree, and each coefficient depends only on the independent variable (x), then the differential equation is linear. 2.Otherwise, the differential equation is nonlinear. V.RAMYA 35 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 36 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 37 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 38 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

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V.RAMYA 40 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 41 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1

V.RAMYA 42 Examples: 1.x−−√+10=26 x 2 −5−−−−−√+x−1