Differential equation topics explanied very good

Differential equation exercise

1. Ordinary Differential Equations Dr. Rashid Mahmood Centre for Mathematical Sciences, PIEAS

2. EXPLICIT AND IMPLICIT SOLUTIONS A solution in which the dependent variable is expressed solely in terms of the independent variable and constants is said to be an explicit solution.

3. Verification of an Implicit Solution

4. FAMILIES OF SOLUTIONS The study of differential equations is similar to that of integral calculus.

5. FAMILIES OF SOLUTIONS(cont.)

6. Particular Solution

7. Examples The one parameter family is an explicit solution of linear first order equation Interval .  

8. Two Parameter Family of Solutions

9. Using Different Symbols

10. A Piecewise-Defined Solution

11. A Piecewise-Defined Solution(cont.)

12. A Piecewise-Defined Solution(cont.)

13. Systems Of Differential Equations

14. Example

15. Example(Cont.)

16. Example 2

17. Initial-value Problems

18. Interval I of Definition of a Solution Consider first-order differential equation And Solution is family of solutions gives,  

19. Example-Second-Order IVP Solution X= C1 cos 4t + C 2 sin 4t