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Section 4.3. UNDAMPED FORCING AND RESONANCE. A trip down memory lane…. Remember section 1.8, Linear Equations? Example: Solve the first-order linear nonhomogeneous equation: The general solution is a sum of The general solution to the associated homogeneous equation

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section 4 3

Section 4.3

UNDAMPED FORCING AND RESONANCE

a trip down memory lane
A trip down memory lane…

Remember section 1.8, Linear Equations?

Example: Solve the first-order linear nonhomogeneous equation:

The general solution is a sum of

  • The general solution to the associated homogeneous equation
  • A particular solution to the given nonhomogeneous equation
the solution
The solution

The general solution to

is

Explain where each term in the solution comes from. What methods were used to obtain the solution?

an unexpected wrinkle
An unexpected wrinkle…

Here’s a new equation:

  • How is it different from the previous equation?
  • How is it the same?
  • What’s the name of this kind of DE? Guess.
  • What methods can we use? Guess again!
method
Method

Example: Solve the second-order linear nonhomogeneous equation:

The general solution is a sum of

  • The general solution to the associated homogeneous equation
  • A particular solution to the given nonhomogeneous equation
homogeneous solution
Homogeneous solution

Solve

which is the DE for the _________________________.

The general solution is

undamped harmonic oscillator

one solution to nonhomogeneous de
One solution to nonhomogeneous DE

Remember that we just need one solution to

Guess:

Plug in to DE:

So  = 4/3 and

general solution to original de
General solution to original DE

Putting it all together, the general solution to

is

The solution to the IVP y(0) = 0 and y’(0) = 0 is

exercise
Exercise

p. 419, #11

why these pictures
Why these pictures?

Use BeatsAndResonance and set the parameters a and  to see the graphs of the solution

Experiment with changing the values of a and . You should see that (except in very special cases) the solutions look like the product of trig functions.

Here’s a handy-dandy trig formula:

Now set A = 3t/4 and B = t/4. So

the solution1
The solution

The solution to the IVP

is

This formula is not in the book (they make you sweat it out with complex exponentials).

What happens when a ? When aand  are not close? When a= ? Experiment with the applet.

some answers

envelope of beats

freq = (a - )/2

ampl = 2F0/(a2 - 2)

freq = (a + )/2

ampl = 1

Some answers

Again, the solution to the IVP is

  • As a , the amplitude of the beats  ∞.
  • When aand  are not close, the amplitude is smaller and the beats are not noticeable.
  • When a= , resonance occurs.
exercise1
Exercise
  • Use the formula on the previous slide to do p. 419, #17.
  • Use the guess-and-test method to try to solve p. 419, #13. Why doesn’t it work?
  • Look up the solution to p. 419, #13 in the back of the book. Verify that it is a solution.
resonance
Resonance

What happens when a= ?

We can’t just plug a=  into the formula. However, we can use L’Hospital’s Rule to find the limit of the envelope function as a  .

so a particular solution when a=  is

and the general solution is

exercise2
Exercise
  • Finish p. 419, #13.
  • p. 420, #21