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Mathematics

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Session Differential Equations - 2

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Session Objectives Method of Solution: Separation of Variables Differential Equation of first Order and first Degree Equations Reducible to Variable Separable Form Homogeneous Differential Equations Method of Solution Class Exercise

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Differential Equation of first Order and first Degree where f (x, y) is the function of x and y. A differential equation of the first order and first degree contains independent variable x, dependent variable y and its derivative

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Separation of Variables Differential equation of the form [where C is an arbitrary constant]

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Separation of Variables [where C is an arbitrary constant] Differential equation of the form

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Separation of Variables [where C is an arbitrary constant] Differential equation of the form

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Example - 1

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Solution Cont. Integrating both sides, we get

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Example - 2

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Solution Cont. Integrating both sides, we get

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Example – 3

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Solution Cont. Integrating both sides, we get

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Solution Cont.

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Example - 4 Solution: We have Solve the differential equation cosy dy + cosx siny dx = 0; given that [Integrating both sides]

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Solution Cont.

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Reducible to Variable Separable Form Substitute ax + by + c = v to reducing variable separable form. Differential equation of the form

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Example - 5 Solve the differential equation:

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Solution Cont.

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Homogeneous Function A function f (x, y) in x and y is called a homogenous function, if the degrees of each term are equal. Examples: is a homogenous function of degree 2 is a homogenous function of degree 3

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Homogenous Differential Equations where f (x, y) and g(x, y) is a homogenous functions of the same degree in x and y, then it is called homogenous differential equation. is a homogenous differential equation as and both are homogenous functions of degree 3. Example:

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Method of Solution (2) Substitute and in the equation. (3) The equation reduces to the form (4) Separate the variables of v and x. (5) Integrate both sides to obtain the solution in terms of v and x. (6) Replace v by to get the solution (1) Write the differential equation in the form

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Example – 6 It is a homogeneous differential equation of degree 1.

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Solution Cont. Integrating both sides, we get

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Example - 7 Solve the differential equation: It is a homogeneous differential equation of degree 2.

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Solution [Integrating both sides]

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Solution Cont.

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Thank you

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