1. Chapter 2 Solution of Differential Equations
Separation of Variables
Dr. Khawaja Zafar Elahi

2. Chapter 2

3. Chapter 2

4. Example
Chapter 2

5. Chapter 2

6. Chapter 2

7. Domain
Chapter 2

8. Partial derviavtive with respect to y
Chapter 2

9. EXAMPLE
STEP 1
STEP 2
STEP 3
Chapter 2

10. Solution
STEP 4
STEP 5
STEP 6
Chapter 2

11. STEP 1
STEP 2
STEP 3
STEP 4
Chapter 2

12. Chapter 2

13. Method for solving First Order Differential Equations
Chapter 2

14. Homogeneous differential Equations Exact Differential Equations
Methods for solving First order Differential Equations
Separable Variable
Homogeneous differential Equations
Exact Differential Equations
Making Exact by Integrating Factor
First order Linear Differential Equation
Chapter 2

15. Methods

16. Variable Separable Differential Equation Chapter 2
Separable Variable
x is independent variable and y is dependent variable or
Variable Separable Differential Equation Chapter 2 or
are separable forms of the differential equation
General solution can be solved by directly integrating both the sides
+ c
Where c is constant of integration
DO YOU REMEMBER INTEGRATION FORMULA?
Chapter 2

17. Separation of Variables
A differential equation of the type y’ = f(x)g(y) is separable.
Definition
Example
Variable Separable Differential Equation Chapter 2
Separable differential equations can often be solved with direct integration. This may lead to an equation which defines the solution implicitly rather than directly.
Example

18. Variable Separable Differential Equation Chapter 2
EXAMPLE:
Variable Separable Differential Equation Chapter 2
Chapter 2

19. Variable Separable Differential Equation Chapter 2

20. Variable Separable Differential Equation Chapter 2

21. Note1: If we have Note.2. If we have Note.3. If we have
Integrating by parts
Note If we have
Integrating by parts
Note If we have
Chapter 2

22. Variable Separable Differential Equation Chapter 2

23. Chapter 2

24. Chapter 2

25. Solution: Example: 12 Separable differential equation
Combined constants of integration

26. Example (cont.):
We now have y as an implicit function of x.
We can find y as an explicit function of x by taking the tangent of both sides.