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# LaPlace Transformations with some Initial Value Problems

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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

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https://brainmass.com/math/calculus-and-analysis/laplace-transformations-initial-value-problems-37769

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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 ...

#### 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.

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