Method Homogeneous Equations Reducible to separable.

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Presentation transcript:

Method Homogeneous Equations Reducible to separable

Homogeneous Differential Equation Chapter 2 Homogenous Differential Equations A differential equation Homogenous differential equation if every t, where t  R Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2 Example:1. Show that differential equation is homogenous differential equation. Solution: Differential equation is homogeneous Homogeneous Differential Equation Chapter 2  Differential equation is homogeneous Chapter 2

Homogeneous Differential Equation Chapter 2 METHOD for solving Homogenous differential equations Substitute Homogeneous Differential Equation Chapter 2 OR Substitute Chapter 2

Homogeneous Differential Equation Chapter 2 . Using substitution the homogeneous differential equation is reduce to separable variable form. Example:2 Solve the homogenous differential equation Solution: Homogeneous Differential Equation Chapter 2 Rewriting in the form : and substitute Chapter 2

Homogeneous Differential Equation Chapter 2 is variable separable form is general solution. Chapter 2

Homogeneous Differential Equation Chapter 2 Note. Selection of substitution Differential Equation depends on number of terms of coefficients , then take 1. If Homogeneous Differential Equation Chapter 2 , then take 2. If 3. If , then take x = vy or y = ux Chapter 2

Homogeneous Differential Equation Chapter 2 Example:. Solve the Differential Equation by using appropriate substitution (1 / 2) Solution: Differential equation is homogeneous as degree of each term is same, hence we can use either y = ux or x = vy as substitution Homogeneous Differential Equation Chapter 2 Substituting y and dy in the given equation Chapter 2

Homogeneous Differential Equation Chapter 2 Separating variable u and x (2 / 2) is Separable form Integrating both the sides Homogeneous Differential Equation Chapter 2 is general solution of the differential equation Chapter 2

Homogeneous Differential Equation Chapter 2 Example: Show that differential equation (1 / 2) is homogeneous Solution: Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2 (2 / 2) is Separable form Integrating both the sides Let Homogeneous Differential Equation Chapter 2 is general solution of the differential equation Chapter 2

Homogeneous Differential Equation Chapter 2

Homogeneous Differential Equation Chapter 2 (1 / 3) Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2 (2 / 3) Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2 (3 / 3) Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2

Homogeneous Differential Equation Chapter 2 (1 / 2) Homogeneous Differential Equation Chapter 2 Chapter 2

Homogeneous Differential Equation Chapter 2 (2 / 2) Homogeneous Differential Equation Chapter 2 is general solution of differential equation Chapter 2

Differential Equation Chapter 2

Differential Equation Chapter 2

Differential Equation Chapter 2 is general solution of differential equation Chapter 2