LaPlace Transformations with some Initial Value Problems

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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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.

See attached for full description of problems. ... some circuit, when does the voltage fall to 37% of its initial value? ... What is the Laplace transform of d? / dt ? ...

... a) the thermocouple is shown with an initial temperature the ... 1.56)]= e^(-1.56t) Putting these values in eq ...problem can be found by Laplace transformation of the ...

... x,t). The solution to the initial problem is then ... we must try another method (eg: Laplace transforms or numerical ... series will converge to the average value of f ...

... capacitor is 2 V and the initial current flowing ... Converting the functions to Laplace transforms, we will get: [ ] R 1 ... 0 since it is unspecified value), we will ...

... range of -1 to +1, then they have no unique value. ... is x(t) = cos(t), t ≥ 0 and no initial energy. ... we will convert x(t) to s-domain using Laplace transform: ...

... capacitor is 2 V and the initial current flowing ... Converting the functions to Laplace transforms, we will get: [ ] R 1 ... 0 since it is unspecified value), we will ...

... single value of τ seconds so the equation becomes; ... T is the Laplace transform of ∆T ... T 0 = the temperature difference at initial conditions immediately ...

... function is given by applying inverse Laplace transform and Z ... c) Initial value theorem in Z domain can be extended to W domain by the transformation 1 + wT ...

... 14 and with the values given (J = 1 Kg-m2 and ... In the interval 0 ≤ t < 0.01, the initial condition is zero ... Taking the Laplace transform of the state equations. ...