Y'' - 9y = 4t - δ_2(t); y() = -2, y'(0) = -1

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In summary, the conversation discusses solving a differential equation using Dirac delta function, Heaviside function, Laplace and Inverse Laplace Transforms. The person is seeking help with finding the inverse Laplace Transform of a specific term in the equation. They have shared their work and are asking for assistance in finding the solution. The conversation also mentions using partial fraction decomposition and a formula for dealing with the term in question.
  • #1
s3a
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Homework Statement


Solve the differential equation y'' - 9y = 4t - δ_2(t); y() = -2, y'(0) = -1 for y(t) when 0 <= t < 2 as well as when t > 2.


Homework Equations


Dirac delta function. Heaviside function. Laplace and Inverse Laplace Transforms.


The Attempt at a Solution


I'll type the relevant information here since my handwriting for my work is ugly (but I am uploading it anyways just in case you need it). I get to Y(s) = 4/[s^2 * (s^2 - 9)] - e^(-2s)/(s^2 - 9) - (2s+1)/(s^2 - 9) and I'm stuck at getting the inverse Laplace Transform of the bolded/italicized term. Could someone please show me how to do that part? I know it involves the unit step/Heaviside function but I don't know what to do specifically.

From Wolfram Alpha,
The inverse Laplace Transform of that bolded/italicized term is:
http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+{-exp(-2s)/(s^2+-+9)}
The inverse Laplace Transform of the whole (s) = 4/[s^2 * (s^2 - 9)] - e^(-2s)/(s^2 - 9) - (2s+1)/(s^2 - 9) equality is:
http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+{4/[s^2+*+(s^2+-+9)]+-+exp(-2s)/(s^2+-+9)+-+(2s%2B1)/(s^2+-+9)}

(Just to note, the Wolfram Alpha answers would be wrong if my work leading up to the equation I retyped in this forum post from my work is incorrect.)

Thanks in advance!
 

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  • #2
s3a said:

Homework Statement


Solve the differential equation y'' - 9y = 4t - δ_2(t); y() = -2, y'(0) = -1 for y(t) when 0 <= t < 2 as well as when t > 2.


Homework Equations


Dirac delta function. Heaviside function. Laplace and Inverse Laplace Transforms.


The Attempt at a Solution


I'll type the relevant information here since my handwriting for my work is ugly (but I am uploading it anyways just in case you need it). I get to Y(s) = 4/[s^2 * (s^2 - 9)] - e^(-2s)/(s^2 - 9) - (2s+1)/(s^2 - 9) and I'm stuck at getting the inverse Laplace Transform of the bolded/italicized term. Could someone please show me how to do that part? I know it involves the unit step/Heaviside function but I don't know what to do specifically.

From Wolfram Alpha,
The inverse Laplace Transform of that bolded/italicized term is:
http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+{-exp(-2s)/(s^2+-+9)}
The inverse Laplace Transform of the whole (s) = 4/[s^2 * (s^2 - 9)] - e^(-2s)/(s^2 - 9) - (2s+1)/(s^2 - 9) equality is:
http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+{4/[s^2+*+(s^2+-+9)]+-+exp(-2s)/(s^2+-+9)+-+(2s%2B1)/(s^2+-+9)}

(Just to note, the Wolfram Alpha answers would be wrong if my work leading up to the equation I retyped in this forum post from my work is incorrect.)

Thanks in advance!

Have you tried splitting up 1/(s2 - 9) into two terms using partial fraction decomposition?
 
  • #3
I haven't because I didn't foresee any upcoming successful step.

Here is my work for what you mentioned.
 

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  • #4
That work looks OK.

What you need to deal with the expressions you're asking about is this formula:

[tex]L^{-1}(e^{-as}~G(s)) = u_a g(t - a)[/tex]

Here G(s) = [itex]L {g(t)}[/itex]

Edit: For some reason, the LaTeX isn't rendering correctly. Also, I neglected to indicate that I was talking about the inverse Laplace transform in the first equation above.
 
Last edited:
  • #5
Did you mean:

L{u_c(t) f(t-c)} = e^(-cs) L{f(t)} ?
 
  • #6
s3a said:
Did you mean:

L{u_c(t) f(t-c)} = e^(-cs) L{f(t)} ?

The first formula should have been for the inverse Laplace transform. It's now fixed.
 
  • #7
Wrong thread sorry.
 
Last edited:

Related to Y'' - 9y = 4t - δ_2(t); y() = -2, y'(0) = -1

1. What is the relationship between "y" and "t" in this equation?

The equation "Y'' - 9y = 4t - δ_2(t); y() = -2, y'(0) = -1" represents a second-order differential equation, where "y" is a function of "t" and its derivatives. The equation shows how the second derivative of "y" is related to "y" and "t," and how "y" is affected by the function δ_2(t).

2. How do the initial conditions y(0) = -2 and y'(0) = -1 affect the solution of this equation?

The initial conditions provided in the equation indicate the starting values for "y" and its first derivative at "t=0." These values are essential in solving the differential equation and determining the specific solution for "y."

3. What is the significance of δ_2(t) in this equation?

The function δ_2(t) represents an external force or disturbance acting on the system. It can be any function of "t" and can affect the behavior of "y" in the equation.

4. How do you solve a second-order differential equation like this one?

To solve this type of differential equation, you will need to use techniques such as separation of variables, the method of undetermined coefficients, or the method of variation of parameters. These methods involve finding a particular solution that satisfies the given initial conditions and the general solution of the equation.

5. Can you provide an example of a physical system that this equation could represent?

This equation could represent the motion of a damped harmonic oscillator, where "y" represents the displacement of the oscillator from its equilibrium position, and "t" represents time. The term δ_2(t) could represent an external force, such as friction or air resistance, acting on the oscillator.

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