## 2280 12:55pm Lectures Week 7 S2017

Last Modified: February 24, 2017, 11:19 MST.    Today: September 25, 2018, 22:39 MDT.
```Topics
Sections 7.1 to 7.6
The textbook topics, definitions, examples and theoremsEdwards-Penney Ch 3, 7.1 to 7.5 (21.0 K, txt, 22 Feb 2015)```

### Week 7, Sections 7.1, 7.2, 7.3

#### Monday-Tuesday: Ch 7

```Exam Review: StudySample Exam 1 with solutions (170.3 K, pdf, 09 Feb 2017) for Midterm 1 on Friday, Feb 26.

Laplace Theory
Forward table
Backward table

Basic Theorems of Laplace Theory
Functions of exponential order
Existence theorem for Laplace integrals
Euler solution atoms have a Laplace integral
Forward table
Backward table
Lerch's theorem
Linearity.
The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Shift theorem L(exp(at)f(t)) = L(f(t))|s->(s-a)
Parts theorem L(y')=sL(y)-y(0)
Parts formula derivation.

Solving differential equations by Laplace's method.
Slide: Solving y' = -1, y(0)=2 with Laplace's method
Laplace's method and quadrature for higher order equations and systems
Solving x'' + 4x = t exp(-t), x(0)=1, x'(0)=0 by the Laplace method
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (188.3 K, pdf, 14 Mar 2016) Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016) Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016) Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016) Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016) Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016) Manuscript: DE systems, examples, theory (730.9 K, pdf, 09 Apr 2014) Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016) Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 20 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: Laplace theory 2008-2017 (546.8 K, pdf, 24 Feb 2017) Transparencies: Ch7 Laplace solutions 7.1 to 7.4 (from EP 2250 book) (1068.7 K, pdf, 22 Feb 2015)```

#### Wednesday and Monday: Gamma, Piecewise Functions, Convolution, Resolvent

```DEF: Piecewise Continuous Function
Existence of the Laplace integral.
One-sided and two-sided Laplace integral
Freeway example, suspension collides with a ramp.
DEF. Gamma function
Gamma(t) = integral x=0 to x=infinity  x^{t-1} e^{-x}
Gamma(n)=(n-1)!, generalizes the factorial function
DEF. Mellin transform
{Mf}(s)= phi(s)=integral x=0 to x=infinity  x^{s-1} f(x)
DEF. Two-sided Laplace transform
{Bf}(s) = {Mf(-ln(x))}(s) = integral x=0 to x=infinity x^{s-1}f(-ln x)
DEF. Unit step u(t-a)=1 for t>=a, else zero
DEF. Ramp t->(t-a)u(t-a)
Backward table problems: examples
Forward table problems: examples
Computing Laplace integrals L(f(t)) with rules
Solving an equation L(y(t))=expression in s for y(t)
Partial fraction methods
Trig identities and their use in Laplace calculations
Hyperbolic functions and Laplace calculations
Why the forward and backward tables don't have cosh, sinh entries
```
``` Piecewise Functions
Unit Step: u(t)=1 for t>=0, u(t)=0 for t<0.
Pulse: pulse(t,a,b)=u(t-a)-u(t-b)
Ramp: ramp(t-a)=(t-a)u(t-a)
L(u(t-a)) = (1/s) exp(-as) [for a >= 0 only]
Integral Theorem
L(int(g(x),x=0..t)) = s L(g(t))
Applications to computing ramp(t-a)
L(ramp(t-a)) = (1/s^2) exp(-as) [for a >= 0 only]
Piecewise defined periodic waves
Square wave: f(t)=1 on [0,1), f(t)=-1 on [1,2), 2-periodic
Triangular wave: f(t)=|t| on [-1,1], 2-periodic
Sawtooth wave: f(t)=t on [0,1], 1-periodic
Rectified sine: f(t)=|sin(kt)|
Half-wave rectified sine: f(t)=sin(kt) when positive, else zero.
Parabolic wave
Periodic function theorem
Proof details
Laplace of the square wave. Problem 7.5-25.

Applications of Laplace's method from 7.3, 7.4, 7.5
Convolution theorem
DEF. Convolution of f and g = f*g(t) = integral of f(x)g(t-x) from x=0 to x=t
THEOREM. L(f(t))L(g(t))=L(convolution of f and g)
Application:   L(cos t)L(sin t) = L(0.5 t sin(t))
Second shifting Theorems
e^{-as}L(f(t))=L(f(t-a)u(t-a)) Backward table
L(g(t)u(t-a))=e^{-as}L(g(t+a)) Forward table
EXAMPLES.
Forward table
L(sin(t)u(t-Pi)) = e^{-Pi s} L(sin(t)|t->t+Pi)
= e^{-Pi s} L(sin(t+Pi))
= e^{-Pi s} L(sin(t)cos(Pi)+sin(Pi)cos(t))
= e^{-Pi s} L(-sin(t))
= e^{-Pi s} ( -1/(s^2+1))
Backward table
L(f(t)) = e^{-2s}/s^2
= e^{-2s} L(t)
= L(t u(t)|t->t-2)
= L((t-2)u(t-2))
Therefore f(t) = (t-2)u(t-2) = ramp at t=2.

Laplace Resolvent Method.
==> This method is a shortcut for solving systems by Laplace's method.
==> It is also a convenient way to solve systems with maple.Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)```

#### Friday: Exam 1

Start at 12:50pm and end at 1:50pm.
Exam Prep Materials:
Sample Exam
Sample Exam Solutons