How Does Laplace Transform Solve the Harmonic Motion Differential Equation?

Click For Summary
SUMMARY

The discussion focuses on solving the harmonic motion differential equation ¨x + 2 ˙x + 2x = 6 sin(t)U(t − 3π/2) with initial conditions x(0) = 2 and x˙(0) = 2 using the Laplace transform. Participants emphasize the importance of applying the Laplace transform to convert the differential equation into an algebraic equation, facilitating the solution for the position function x(t). The use of the Heaviside step function U(t − 3π/2) is also highlighted as crucial for handling the non-homogeneous part of the equation.

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear equations.
  • Familiarity with the Laplace transform and its properties.
  • Knowledge of initial value problems and how to apply initial conditions.
  • Concept of the Heaviside step function in piecewise functions.
NEXT STEPS
  • Study the properties of the Laplace transform, including linearity and shifting theorems.
  • Learn how to apply the inverse Laplace transform to retrieve time-domain solutions.
  • Explore the application of the Heaviside step function in solving differential equations.
  • Investigate the method of undetermined coefficients for solving non-homogeneous differential equations.
USEFUL FOR

Students and educators in mathematics, particularly those studying differential equations, as well as engineers and physicists dealing with harmonic motion and system dynamics.

Jim wah
Messages
6
Reaction score
0
(2.) Let the differential equation ¨x + 2 ˙x + 2x = 6 sin(t)U(t − 3π/2) , x(0) = 2, x˙(0) = 2

Solve for the position function x(t) using the Laplace transform:
 
Physics news on Phys.org
Hi Jim,

Use the template provided for such things. It helps you order your thoughts and you get much better assistance.:

Homework Statement


obvious​

Homework Equations


what can you mobilize to deal with this ? notes, textbook, ...​

The Attempt at a Solution


Solution of the homogeneous equation is ...
what does Laplace do with equations like this ?​
 
Jim wah said:
Solve for the position function x(t) using the Laplace transform:
 

Similar threads

Undergrad Laplace transform of a simple equation (Simple question)
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Applying the Laplace transform to solve Differential equations
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Solving a differential equation using Laplace transform
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad On Laplace transform of derivative
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Solution of delayed forcing function
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Convergence of this Laplace transformation
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Rational Chebyshev Collocation Method For Damped Harmonic Oscilator
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Transforming Cartesian Coordinates in terms of Spherical Harmonics
  • · Replies 1 ·
Replies
1
Views
4K
MHB Mahesh's question via email about Laplace Transforms (1)
  • · Replies 1 ·
Replies
1
Views
10K
Undergrad On a bijective Laplace transform
  • · Replies 1 ·
Replies
1
Views
3K