Mathematics

Session Differential Equations - 2

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

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

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

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

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

Example - 1

Solution Cont. Integrating both sides, we get

Example - 2

Solution Cont. Integrating both sides, we get

Example – 3

Solution Cont. Integrating both sides, we get

Solution Cont.

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

Solution Cont.

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

Example - 5 Solve the differential equation:

Solution Cont.

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

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:

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

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

Solution Cont. Integrating both sides, we get

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

Solution [Integrating both sides]

Solution Cont.

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