Finding Solutions to DE using Laplace Transform

In summary, the conversation discusses solving a differential equation with a given function and using Heaviside functions to transform it. The solution involves using u1 and u3, but the final answer only includes u1. However, additional information is needed, such as initial conditions, to fully solve the equation.
  • #1
manenbu
103
0

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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  • #2
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?
 
  • #3
it doesn't look as though you changed anything by putting it in unit step form.
 
  • #4
Sorry, forgot.
y(0) = 1, y'(0) = 2.
 

1. What is Laplace transform and how is it used in differential equations?

Laplace transform is a mathematical tool used to solve differential equations. It converts a function of time into a function of complex frequency. This transformation allows for easier manipulation and solution of differential equations.

2. Can Laplace transform be used for all types of differential equations?

No, Laplace transform is most commonly used for linear differential equations with constant coefficients. It may also be used for some non-linear or time-varying equations, but it is not applicable for all types of differential equations.

3. How does Laplace transform handle initial conditions?

Laplace transform incorporates initial conditions into the solution by using the Heaviside step function. The initial conditions are applied to the transformed equation, and then the inverse Laplace transform is taken to find the solution in the time domain.

4. What are the advantages of using Laplace transform to solve differential equations?

Laplace transform can simplify the process of solving differential equations by transforming them into algebraic equations, making them easier to manipulate. It also allows for the use of tables and properties to solve equations, rather than tedious integration methods.

5. Are there any limitations to using Laplace transform for solving differential equations?

Yes, there are some limitations to using Laplace transform. It may not be applicable for all types of differential equations, as mentioned earlier. It also requires the original function to be defined for all positive time values, which may not always be the case in real-world applications.

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