Purchase Solution

# LaPlace Transformations with some Initial Value Problems

Not what you're looking for?

Problem 9.1 (Prob. 29. P. 252) Two particles each of mass m moves in the plane with co-ordinates (x(t), y(t)) under the influence of a force that is directed toward the origin and had magnitude k/(x2 + y2) an inverse-square central force field. Show that mx''=-kx/(r^3) and my''= -ky/(r^3) where r = sqrt(x2 + y2)

Problem 9.2 (Prob. 30, P. 252) suppose that a projectile of mass in moves in a vertical plane in the atmosphere near the surface of the earth under the influence of two forces: a downward gravitational force of magnitude mg and a resistive force FR that is directed opposite to the velocity vector v and has magnitude kv^2 (where v = |v| is the speed of the projectile). Show that the equations of motion of the projectile are
mx" = ?kvx'
my" =?kvy"?mg

Problem 9.3 Use Laplace Transfonn to find the particular solutions of the following systems
x' = ?y
y'=13x+4y
x(0) = 0
y(0) = 3

Please see attached for the rest of this question, and all other questions.

##### Solution Summary

Eight problems involving Laplace Transformations and Initial-Value Problems are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

##### Solution Preview

Hello and thank you for posting your question to Brainmass!
The solution is attached below in two Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.

Check out:
http://mathworld.wolfram.com/LaplaceTransform.html
http://mathworld.wolfram.com/Convolution.html
http://sosmath.com/diffeq/laplace/basic/basic.html

The acceleration is defined as the second derivative of the displacement with respect to time:

The acceleration according to Newton's second law is:

The last equation is a vector equation and we have to find the force components along the x and the y axis.

Defining:

Let q be the angle between the force and the positive x-axis. Therefore:

In polar coordinates, the position of the particle is:

Hence:

So we get:

But:

Thus:

As in the previous problem, we have to write the forces components at each direction and equate them to the second derivative of the displacement along the respective axis.

In the x direction ...

##### Free BrainMass Quizzes

This quiz test you on how well you are familiar with solving quadratic inequalities.

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.