Ordinary differential equations of the first order and methods of their solutions

1. Mathematical Analysis 2

Lecture 3 - Ordinary differential equations of the
first order and methods of their solutions.
Akbota Myrzakul
a.myrzakul@astanait.edu.kz
Astana IT University

2. Lecture overview

• Basic definitions of ODE
• Classify ODE based on( order, linearity, conditions)
• Linear first order ODE

3. Basic definitions of ODE

4. Basic definitions of ODE

5. Ordinary Differential Equations (ODE)

6. Order of a differential equation

7. Solution of a differential equation

8. Linear ODE

9. Boundary-value and Initial value problems

10. Classification of ODE

ODE can be classified in different ways
• Order
– First order ODE
– Second order ODE
– Nth order ODE
• Linearity
– Linear ODE
– Nonlinear ODE
• Auxiliary conditions
– Initial value problems
– Boundary value problems

11. Linear first order ODE

12. Solution of Linear first order ODE

13. Solution of Linear first order ODE

14. Example

15. Example

16. Solution of Linear first order ODE

17. Separable equation

18. Homogeneous Equations

19. Homogeneous Equations

20. Homogeneous Equations

21. Homogeneous Equations

22. Homogeneous Equations

23. Homogeneous Equations

24. Homogeneous Equations

25. Linear first order ODE

26. Solution of Linear first order ODE

27. Solution of Linear first order ODE

28. Solution of Linear first order ODE

29. Summary of the Solution Process

30. Example

31. Example

xy' 2 y x x, y(1)
2
1
2
y'
2
y x 1
x
p ( x ) dx
p ( x ) dx
y(x) e
e
g
(
x)d
x
c
2
2
dx
dx
2
2
x
x
e
(
x
1)d
x
c
e
x x ( x 1)d x c
1
x 2 x 4 1 x 3 c 1 x 2 1 x cx 2
3
3
4
4
Apply the initial c o nditio n to ge t c,
1 1 1
7
c c
.
2 4 3
12

32. Lecture Summary

• Basic definitions of ODE
• Classify ODE based on( order, linearity, conditions)
• Linear first order ODE