Solving the differential equations involving SHM

In summary, when dealing with solving differential equations for damped and driven simple harmonic motion, the most satisfactory explanation is to guess certain solutions and then check if they work. This approach is based on past experience and often involves looking for the simplest function that fits the rules, such as a single frequency sine wave. However, when solving equations for waves, there are multiple possibilities and it is important to remember that a sine wave is not the only solution.
  • #1
anirocks11
What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?
 
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  • #2
It is easy to check if a solution works once you make a guess. So generally you just have experience with non damped systems, and you guess and check until you get the form that works.
 
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  • #3
anirocks11 said:
guessing certain solutions
That is a real problem when dealing with 'the next step' in Maths, for all students. "Why did they choose to do it that way?" we ask. Finding the way to the next step is always based on past experience so a student can't expect to find it to be very obvious.
In the SHM case, 'they' looked amongst the simplest function that would fit the rules - i.e. repeating and continuous etc etc. A single frequency sine wave is the first thing that would come to mind for SHM and it happens to be the right solution. When trying to solve the equations for Waves, there are many more possibilities (square waves, sawtooth etc) and they also work but we tend to start with a sine wave again - but we have to remember it's not the only one in that case.
 

1. What is a differential equation?

A differential equation is an equation that relates a function to its derivatives. It describes the relationship between a dependent variable and its independent variables, and is used to model many natural phenomena.

2. What is SHM?

SHM stands for Simple Harmonic Motion. It is a type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium. Examples of SHM include a mass-spring system and a pendulum.

3. How do you solve a differential equation involving SHM?

To solve a differential equation involving SHM, you first need to identify the equation that describes the motion. This is usually a second-order differential equation. Then, you can use mathematical techniques such as separation of variables, substitution, or Laplace transforms to solve the equation and find the solution for the motion.

4. What are the key concepts to understand when solving differential equations involving SHM?

The key concepts to understand when solving differential equations involving SHM include the definition of SHM, the properties of periodic motion, the relationship between force and displacement, and the mathematical techniques used to solve differential equations.

5. What are some real-life applications of solving differential equations involving SHM?

Solving differential equations involving SHM has many real-life applications, such as in the design of springs for car suspensions, understanding the behavior of a swinging pendulum, and studying the motion of molecules in a vibrating chemical bond. It is also used in fields such as engineering, physics, and biology to model and analyze various systems and phenomena.

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