Advanced Engineering Mathematics by Erwin Kreyszig Copyright  2007 John Wiley & Sons, Inc. All rights reserved. Engineering Mathematics Lecture 05: 2.

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Presentation transcript:

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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