1. Method
Homogeneous Equations
Reducible to separable

2. 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

3. 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

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

5. 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

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

7. 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

8. 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

9. 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

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

11. 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

12. Homogeneous Differential Equation Chapter 2

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

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

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

16. Homogeneous Differential Equation Chapter 2

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

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

19. Differential Equation Chapter 2

20. Differential Equation Chapter 2

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