1. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Engineering Mathematics Lecture 05: 2 nd order ODEs (Cont’d) In Last Meeting  Homogeneous Linear ODE (2 nd order)  Two distinct real roots: Exponential decay/increase  Real double root: (a+bx)*exp( x)  Reduction by integrating factor  Complex conjugate roots: Oscillatory behavior  Harmonic Oscillation with Spring and Weight  Real/Complex Roots : Over-/Under-damping, respectively.  Today  Non-homogeneous ODE  More examples in Mechanics and Electronics

2. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued Non-Homogeneous Linear ODEs

3. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

4. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

5. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued Method of Undetermined Coefficients

6. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

7. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

8. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued Example

9. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Pages 81-82b

10. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued Modeling: Resonance Damped oscillation Harmonic: c=0, Critical damping : c = sqrt(4mk)

11. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

12. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

13. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

14. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. I. Undamped Forced Oscillation c = 0  b = 0

15. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

16. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

17. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

18. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

19. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

20. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

21. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

22. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

23. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

24. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Modeling: Electrical Circuits

25. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

26. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

27. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

28. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

29. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

30. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

31. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

32. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

33. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

34. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. More Complicated Equations

35. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Continued

36. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved.

37. Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Summary of Chapter 2  Homogeneous Linear ODE (2 nd order)  Two distinct real roots: Exponential decay/increase  Real double root: (a+bx)*exp( x)  Reduction by integrating factor  Complex conjugate roots: Oscillatory behavior  Harmonic Oscillation with Spring and Weight  Real/Complex Roots : Over-/Under-damping, respectively.  Non-homogeneous ODE  General solution Homo. + Particular solution for NHomo.  Most Popular models in Mechanics and Electric Problems