Finding Solutions to DE using Laplace Transform

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Homework Help Overview

The problem involves solving a second-order differential equation using the Laplace transform, with a piecewise function as the forcing term. The original poster is trying to understand the transformation and the resulting solution, particularly the absence of a term in their final answer.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply the Laplace transform to the differential equation and convert the forcing function into Heaviside step functions. They express confusion about the disappearance of a term in their solution and question the presence of another term.
  • Some participants question whether additional initial conditions were provided, which could affect the solution.
  • Others suggest that the transformation into unit step form may not have changed the equation significantly.

Discussion Status

The discussion is ongoing, with participants exploring the implications of the initial conditions and the transformation process. There is an acknowledgment of missing information that could clarify the situation, and various interpretations of the problem are being considered.

Contextual Notes

The original poster has provided initial conditions for the differential equation, which may influence the solution but were initially omitted. The piecewise nature of the forcing function is also a key aspect under discussion.

manenbu
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Homework Statement



y'' -6y' + 9y = f(t)

f(t)=
0, 0<t<1
1, 1<t<3
0, t>3

Homework Equations





The Attempt at a Solution



Turning it to heaviside functions I get:
y'' -6y' + 9y = u1 - u3

and I solve.

in the answers it should be:
y(t) = 2tet + u1(1/9 - 1/9e3(t-1)+ 1/3(t-1)e3(t-1)).

where did u3 go in the answers? why doesn't it appear? the u1 I can get just like that, but I can't seem to figure out where the u3 went and why there is a 2tet in there.
 
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It looks like we're missing some information. Were there additional conditions given? (e.g. y(0) = ? and y'(0) = ?)

Also, what did the DE look like after you applied the Laplace transform?
 
it doesn't look as though you changed anything by putting it in unit step form.
 
Sorry, forgot.
y(0) = 1, y'(0) = 2.
 

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