Google Search in:

## 2280 8:05am Lectures Week 6 S2018

Last Modified: February 20, 2018, 11:22 MST.    Today: September 24, 2018, 11:12 MDT.
```Topics
Sections 3.1 to 3.6
The textbook topics, definitions, examples and theoremsEdwards-Penney Ch 3, 3.1 to 3.4 (16.5 K, txt, 04 Jan 2015)Edwards-Penney Ch 3, 3.5 to 3.7 (17.7 K, txt, 02 Jan 2015)```

#### Monday-Tuesday

```Slides:
Shock-less auto.
Rolling wheel on a spring.
Swinging rod.
Mechanical watch.
Bike trailer.
Physical pendulum.
Classification: critically, over and under damped
Phase-amplitude conversions
Trig right triangle formulas
Triu identities: sin(a+b), others by tricks, even-odd
Cafe door.
Pet door.
Beats.
Undetermined coefficients.
Resonance.

Chapter 3 references. Sections 3.4, 3.5, 3.6. Forced/Unforced oscillations.Slides: Resonance and undetermined coefficients, cafe door, pet door, phase-amplitude (178.0 K, pdf, 08 Mar 2014)Slides: Electrical circuits (112.8 K, pdf, 19 Feb 2016)Slides: Unforced vibrations 2008 (647.6 K, pdf, 27 Feb 2014)Slides: Forced damped vibrations (263.9 K, pdf, 10 Feb 2016)Slides: Forced vibrations and resonance, Millenium Bridge, Wine Glass, Tacoma Narrows (253.0 K, pdf, 08 Mar 2014)Slides: Forced undamped vibrations (214.2 K, pdf, 03 Mar 2012) Slides: phase-amplitude, cafe door, pet door, damping classification (136.0 K, pdf, 08 Mar 2014)

```

#### Tue-Wed: Undetermined Coefficients. Sections 4.1,3.5

```    REVIEW: Undetermined Coefficients
Which equations can be solved
THEOREM. Solution y_h(x) is a linear combination of atoms.
THEOREM. Solution y_p(x) is a linear combination of atoms.
THEOREM. (superposition)  y = y_h + y_pSlides: Basic undetermined coefficients (2018) (147.6 K, pdf, 14 Feb 2018)Slides: Variation of parameters (164.5 K, pdf, 03 Mar 2012)
EXAMPLE. How to find a shortest expression for y_p(x) using
Details for x''(t)+x(t) = 1+t
the trial solution x(t)=A+Bt
the answer x_p(t)=1+t.
BASIC METHOD. Given a trial solution with undetermined coefficients,
find a system of equations for d1, d2, ... and solve it.
Report y_p as the trial solution with substituted
answers d1, d2, d3, ...
THEORY. y = y_h + y_p, and each is a linear combination of atoms.

How to find the homogeneous solution y_h(x) from the characteristic equation.
How to determine the form of the shortest trial solution for y_p(x)
METHOD. A rule for finding y_p(x) from f(x) and the DE.
Finding a trial solution with fewest symbols.
Rule I. Assume the right side f(x) of the differential equation
is a linear combination of atoms. Make a list of all distinct atoms
that appear in the derivatives f(x), f'(x), f''(x), ... . Multiply
these k atoms by undetermined coefficients d_1, ... , d_k, then
add to define a trial solution y.

This rule FAILS if one or more of the k atoms is a solution of
the homogeneous differential equation.

Rule II. If Rule I FAILS, then break the k atoms into groups
with the same base atom. Cycle through the groups, replacing atoms
as follows. If the first atom in the group is a solution of the homogeneous
differential equation, then multiply all atoms in the group by factor x. Repeat
until the first atom is not a solution of the homogeneous differential equation.
Multiply the constructed k atoms by symbols d_1, ... , d_k and add to define trial solution y.

Explanation: The relation between the Rule I + II trial solution and
the book's table that uses the mystery factor x^s.
EXAMPLES.
y'' = x
y'' + y = x exp(x)
y'' - y = x exp(x)
y'' + y = cos(x)
y''' + y'' = 3x + 4 exp(-x)

THEOREM. Suppose a list of k atoms is generated from the
atoms in f(x), using Rule I. Then the shortest trial
solution has exactly k atoms.

EXAMPLE. How to find a shortest trial solution using
Rules I and II.

Details for x''(t)+x(t) = t^2 + cos(t), obtaining
the shortest trial solution
x(t)=d1+d2 t+d3 t^2+d4 t cos(t) + d5 t sin(t).
How to use dsolve() in maple to check the answer.
EXAMPLE. Suppose the DE has order n=4 and the homogeneous
equation has solution atoms cos(t), t cos(t), sin(t),
t sin(t). Assume f(t) = t^2 + cos(t). What is the
shortest trial solution?
EXAMPLE. Suppose the DE has order n=2 and the homogeneous
equation has solution atoms cos(t), sin(t). Assume
f(t) = t^2 + t cos(t).
What is the shortest trial solution?
EXAMPLE. Suppose the DE has order n=4 and the homogeneous
equation has solution atoms 1, t, cos(t), sin(t).
Assume
f(t) = t^2 + t cos(t).
What is the shortest trial solution?
```