Solving DE with Laplace Transforms and g(t) Function

In summary, the conversation discusses solving a differential equation using Laplace transforms, with a piecewise function attached. The work provided is incomplete and misuses formulas. It is suggested to use the definition instead and do both integrals. The unit step function is also mentioned as a correction for a previous typo.
  • #1
s3a
818
8

Homework Statement


Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.


Homework Equations


Laplace transforms of regular and unit step/Heaviside functions.


The Attempt at a Solution


My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!
 

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  • #2


s3a said:

Homework Statement


Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.


Homework Equations


Laplace transforms of regular and unit step/Heaviside functions.


The Attempt at a Solution


My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!

I didn't check all your steps, but it looks to me you haven't taken into account the f(t) = t part. Your formula for that is$$
f(t) = t(1-u(8\pi))+8\pi u(t-8\pi)$$
 
  • #3


What does u(8π) mean?
 
  • #4


s3a said:

Homework Statement


Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.

Homework Equations


Laplace transforms of regular and unit step/Heaviside functions.

The Attempt at a Solution


My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!

Why don't you just use the definition, rather than applying formulas that can be misused (as you did)? For [itex]g(t) = \min(t,8 \pi)[/itex] we have
[tex] L[g](s) = \int_0^\infty e^{-st} g(t) \, dt = \int_0^{8 \pi} e^{-st} t \, dt
+ \int_{8 \pi}^\infty e^{-st} 8 \pi \, dt, [/tex] and just do both integrals.

RGV
 
Last edited:
  • #5


s3a said:
What does u(8π) mean?

Sorry, that was a typo; that expression should be ##u(t-8\pi)## but the time for me to edit and correct it has expired.
 
  • #6


u(t-8∏) is a unit step function. You can think of it as energy going into your system after 8∏ time has elapsed. The function is 0 <= 8∏ and 1 after 8∏.
 

FAQ: Solving DE with Laplace Transforms and g(t) Function

What is a Laplace Transform and how is it used in solving differential equations?

A Laplace Transform is a mathematical tool used to transform a differential equation from the time domain to the frequency domain. This allows us to solve the differential equation using algebraic equations, which are often easier to solve than differential equations. The g(t) function represents the input or forcing function in the differential equation, and is also transformed to the frequency domain.

Can all differential equations be solved using Laplace Transforms and g(t) function?

No, not all differential equations can be solved using Laplace Transforms. The equation must have a linear constant coefficient, meaning that the highest order of the derivative is multiplied by a constant, and the equation must also have initial conditions specified. Additionally, the g(t) function must be known or able to be determined.

How do you find the inverse Laplace Transform to get the solution in the time domain?

To find the inverse Laplace Transform, we use a table of Laplace Transform pairs and their corresponding inverse transforms. We transform the equation to the frequency domain, manipulate it to be in a form that matches one of the pairs in the table, and then use the inverse transform to get the solution in the time domain.

Can Laplace Transforms be used for partial differential equations?

No, Laplace Transforms are only applicable to ordinary differential equations. For partial differential equations, other methods such as separation of variables or numerical methods must be used.

Are there any limitations or drawbacks to using Laplace Transforms in solving differential equations?

One limitation is that Laplace Transforms cannot be used for equations with time-varying coefficients. Additionally, the transformation process can be complex and time-consuming, especially for more complicated equations. It also assumes linearity, which may not always be the case in real-world applications.

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