IVP Laplace Transform Problem - Tricky Inverse Laplace Transform

In summary, the problem involves solving x"+x'+x=1 with given initial values, and using Laplace transforms to simplify the equation to Y(s)=1/((s2+s+1)*s). After using partial fractions, the inverse Laplace transform for the second term can be found by completing the square and breaking it up, resulting in 1-e-t/2cos3t/4-√3/3 * e-t/2sin√3/2.
  • #1
jimagnus
4
0
IVP Laplace Transform Problem -- Tricky Inverse Laplace Transform

Homework Statement



Solve x"+x'+x=1, given x(0)=x'(0)=0


Homework Equations





The Attempt at a Solution



Plugged in transforms: s2*Y(s)-s*y(0)-y'(0)+s*Y(s)-y(0)+Y(s)=1/s
Plugged in initial value points, simplified to Y(s)=1/((s2+s+1)*s)
Partial Fractions led me to Y(s)=1/s+ (-s-1)/(s^2+s+1)

I get stuck at finding an inverse transform for the second term. You can't complete the square for the denominator, right?
 
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  • #2


You can still do a partial fraction expansion even if the roots of the denominator aren't real.
 
Last edited:
  • #3


jimagnus said:
I get stuck at finding an inverse transform for the second term. You can't complete the square for the denominator, right?
Why not?
 
  • #4


If I complete the square, I get the denominator is (s+1/2)2+3/4, and the numerator is -(s+1), and I don't know what inverse laplace transform this would be.
 
  • #5


Hint: s+1 = (s+1/2) + 1/2
 
  • #6


oh right, I didn't think to break it up like that.

so you end up with 1/s-[(s+1/2)/((s+1/2)2+3/4)+1/2 / ((s+1/2)2+3/4), which becomes 1-e-t/2cos3t/4-√3/3 * e-t/2sin√3/2

thanks!
 

1. What is an IVP Laplace Transform Problem?

An IVP (Initial Value Problem) Laplace Transform Problem is a mathematical problem that involves finding the inverse Laplace transform of a function given its Laplace transform and initial conditions. This is a common problem in the field of engineering and physics, where Laplace transforms are used to solve differential equations.

2. What makes the IVP Laplace Transform Problem tricky?

The IVP Laplace Transform Problem can be tricky because it involves multiple steps and sometimes requires clever manipulation of equations to find the solution. It also requires a good understanding of Laplace transforms and their properties.

3. How do you approach solving an IVP Laplace Transform Problem?

The first step is to take the Laplace transform of the given function and use it to form a new equation. Then, apply any necessary algebraic manipulations to isolate the inverse Laplace transform. Finally, use tables or known Laplace transform pairs to find the inverse transform. It is important to always check the initial conditions and ensure they are satisfied in the final solution.

4. Can you provide an example of a tricky IVP Laplace Transform Problem?

One example of a tricky IVP Laplace Transform Problem is solving the differential equation y'' - 2y' + y = 2t with initial conditions y(0) = 0 and y'(0) = 1. The solution involves using partial fractions and inverse Laplace transforms to find the final solution y(t) = t + e^t -1.

5. How are IVP Laplace Transform Problems used in real-world applications?

IVP Laplace Transform Problems are commonly used in engineering and physics for modeling and solving differential equations. They are also used in control systems, signal processing, and circuit analysis. By using Laplace transforms, complex problems can be simplified and solved efficiently, making them a valuable tool in many real-world applications.

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