Laplace Transform of 2nd Order ODE

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SUMMARY

The discussion focuses on solving a second-order ordinary differential equation (ODE) using the Laplace transform. The participant initially struggles with a minor error in their calculations, specifically a misidentified coefficient in their work. They successfully identify that the fifth term in their equation should be "2" instead of "1," leading to the correct application of the Laplace transform. This highlights the importance of careful attention to detail in mathematical problem-solving.

PREREQUISITES
  • Understanding of Laplace transforms, specifically for second-order ODEs.
  • Familiarity with initial value problems in differential equations.
  • Knowledge of the properties of derivatives and their transforms.
  • Ability to manipulate algebraic expressions and fractions accurately.
NEXT STEPS
  • Study the properties of the Laplace transform for higher-order derivatives.
  • Practice solving initial value problems using the Laplace transform method.
  • Review common mistakes in algebraic manipulation related to differential equations.
  • Explore tables of Laplace transforms for various functions, including trigonometric functions.
USEFUL FOR

Students studying differential equations, mathematicians working with Laplace transforms, and educators teaching ODEs who seek to understand common pitfalls in solving these problems.

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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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Nevermind, I figured it out; on the fourth line, the 5th term should be "2" and not "1"
 

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