November 10 Math 2306 sec. 52 Fall 2021

November 10 Math 2306 sec. 52 Fall 2021

Section 16: Laplace Transforms of Derivatives and IVPs

Use the Laplace transform to solve the system of equations

x (t) = y , x(0) = 1, x (0) = 0 y (t) = x, y(0) = 1

We took the transform and used Crammer's Rule to get to the solution

2/3 1/3(s - 1) X (s) = s - 1 + s2 + s + 1

2/3 1/3(s + 2) Y (s) = s - 1 + s2 + s + 1

The irreducible quadratic denominator

s2 + s + 1 =

12 s+ +

2 3 .

2

2

November 8, 2021 1 / 57

2/3

1/3(s - 1)

X (s)

=

+

s-1

s

+

1 2

2+

3

2

2

2/3

1/3(s + 2)

Y (s)

=

+

s-1

s

+

1 2

2+

3

2

2

November 8, 2021 2 / 57

November 8, 2021 3 / 57

November 8, 2021 4 / 57

Section 17: Fourier Series: Trigonometric Series

Consider the following problem:

An undamped spring mass system has a mass of 2 kg attached to a spring with spring constant 128 N/m. The mass is driven by an external force f (t) = 2t for -1 < t < 1 that is 2-periodic so that f (t + 2) = f (t) for all t > 0.

d2x Figure: 2 dt2 + 128x = f (t)

November 8, 2021 5 / 57

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download