Second order ODE into a system of first order ODEs

In summary, the harmonic oscillator's equation of motion is a second-order differential equation with a forcing term of the form f(t) = f0sin(ωt). In order to code it into python and plot the results, it needs to be converted to a system of first-order ODEs. However, the function f(t) should not be removed from the equations in this process. Additionally, initial conditions should also be taken into account.
  • #1
whatisgoingon
9
0

Homework Statement


The harmonic oscillator's equation of motion is:

x'' + 2βx' + ω02x = f

with the forcing of the form f(t) = f0sin(ωt)

The Attempt at a Solution



So I got:
X1 = x
X1' = x' = X2
X2 = x'
X2' = x''
∴ X2' = -2βX2 - ω02X1 + sin(ωt)

The function f(t) is making me doubt this answer because I have to take into account f0 and it just disappears in the solution.

(For context I have to put it into a system of first order ODEs because I have to code it into python and plot the results with the given parameters.)

Am I on the right track? Or am I missing anything?
 
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  • #2
whatisgoingon said:
it just disappears in the solution
Looks to me you made it disappear in the equations already.
 
  • #3
BvU said:
Looks to me you made it disappear in the equations already.
Oh so I shouldn't have taken it out in the first place then? That makes sense thanks! Other than that, am I missing anything else?
 
  • #4
Initial conditions ?
 

1. What is a "second order ODE"?

A second order ODE (ordinary differential equation) is a mathematical equation that involves a function and its derivatives up to the second order. This means that the equation includes the first and second derivatives of the function, and the goal is to find the function that satisfies the equation.

2. How can a second order ODE be converted into a system of first order ODEs?

A second order ODE can be transformed into a system of first order ODEs by introducing new variables and rewriting the equation in terms of these variables. This creates a system of equations where the first order derivatives of the original function are the new variables.

3. Why would someone want to convert a second order ODE into a system of first order ODEs?

Converting a second order ODE into a system of first order ODEs can make it easier to solve the equation using numerical methods or computer algorithms. It can also provide a better understanding of the behavior of the system and make it easier to analyze and interpret the results.

4. What are the benefits of using a system of first order ODEs instead of a second order ODE?

Using a system of first order ODEs can simplify the problem and make it easier to find a solution. It also allows for more flexibility in choosing numerical methods and can provide a better understanding of the underlying dynamics of the system.

5. Are there any limitations to converting a second order ODE into a system of first order ODEs?

Converting a second order ODE into a system of first order ODEs may not always be possible, as some equations may not have a suitable form to be transformed in this way. It also may not always be necessary to convert the equation if a solution can be easily found using other methods.

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