Laplace Transform of 2nd Order ODE

In summary, the conversation is about a problem from a book that requires finding the Laplace transform of an initial value problem. The person is close to finding the solution but thinks they may have made a small error due to being hungry and sleep-deprived. They mention the relevant equations and ask for help. Later on, they realize their mistake and correct it.
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Homework Statement



This is a problem from my book that I'm very close to finding the solution to, but I'm a little off. I have a feeling it's some small error I'm just overlooking because I'm so hungry/sleep-deprived. Anyway, the question asks you to find the Laplace transform of the given initial value problem. I've embedded my work.

Homework Equations



The Laplace transform of first and second order derivatives and of cos(bt) and sin(bt) which are given in the book and I didn't type out because there are tables everywhere.

The Attempt at a Solution



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Thanks for any insight any of you might be able to give. Like I said, I feel like it's just something trivial, some negative sign, or some incorrect distribution of fractions that I'm overlooking.
 
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  • #2
Nevermind, I figured it out; on the fourth line, the 5th term should be "2" and not "1"
 

What is a Laplace Transform?

A Laplace Transform is a mathematical operation that transforms a function from the time domain to the frequency domain. It is commonly used in engineering and physics to solve differential equations.

How is a Laplace Transform used to solve 2nd order ODEs?

In order to solve a 2nd order Ordinary Differential Equation (ODE), the equation is first transformed using a Laplace Transform. This results in a simpler algebraic equation in the frequency domain, which can then be solved using standard methods. Once the solution is found, it is then transformed back to the time domain to get the final solution for the original ODE.

What is the Laplace Transform of a 2nd order ODE with initial conditions?

The Laplace Transform of a 2nd order ODE with initial conditions involves taking the Laplace Transform of both sides of the equation, applying the initial conditions, and then solving for the transformed function. This will result in an equation in the frequency domain that can be solved for the transformed function.

What are the advantages of using Laplace Transforms to solve 2nd order ODEs?

Laplace Transforms have several advantages when solving 2nd order ODEs. They can simplify the equations into algebraic equations in the frequency domain, which can be easier to solve. They also allow for the use of initial conditions, which can be difficult to incorporate in other methods of solving ODEs.

Are there any limitations to using Laplace Transforms to solve 2nd order ODEs?

While Laplace Transforms can be a useful tool in solving 2nd order ODEs, they may not always be applicable. In some cases, the inverse Laplace Transform may not exist, making it impossible to transform back to the time domain. Additionally, certain types of ODEs may not be easily transformed using this method.

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