# NEED: Nonlinear ODE with harmonic output

• X89codered89X
In summary: Your Name]In summary, the conversation was about finding a differential equation that would produce a Fourier series with a fundamental frequency input and sound "warm". The operator L was defined as -a^2x(t) + b^2x''(t), and by plugging in the input cos({\omega}t) and solving for x''(t), it was determined that the equation would have a forcing function of cos({\omega}t) and produce harmonics of that input. By adjusting the constants a and b, the output can be made to sound warmer.
X89codered89X
Hi,

I'm trying to find a differential equation (first OR 2nd order) whose output could be a Fourier Series of a fundamental frequency input... So It will be a "forced" ODE.

That is, I want
$cos({\omega}t) = L[x(t)]$ where
$L$ is some operator on $x(t)$ such that the solutions in $x(t)$ are harmonics of $cos({\omega}t)$ as
$x(t) = a_1cos({\omega}t) + a_2cos(2{\omega}t) + a_3cos(3{\omega}t) + a_4cos(4{\omega}t) +... = \sum_{n = 1}^{\infty}a_ncos(n{\omega}t)$
...and the trick is...I WANT IT TO SOUND "WARM".

MORE ON MY GOAL: I'm actually in my senior design course in engineering and my team is making a synthesizer for our project and we need a way to make warm sounds (e.g. sounds with timbres comfortable to the human ear.) this requires making output that sounds more interesting than simple sinusoids which are easy to generate with a simple linear difference equation (discrete, recursive definition of ODE).

So far, I've tried various things like the forced Duffing equation and the pendulum equation (which are really close to each other actually) , but nothing seems to give predictable output...

Would there be any set of operators that would allow me to change a few constants, say... $a$ and $b$ in $L$ to sweep across a range of neat sounds of $x(t)$?

Last edited:

Hello,

Thank you for reaching out to me with your question. I can definitely help you find a differential equation that meets your requirements.

First, let's define the operator L as follows:

L[x(t)] = -a^2x(t) + b^2x''(t)

Where a and b are constants that can be adjusted to change the output of the equation.

Now, let's plug in our input cos({\omega}t) into this equation:

-a^2cos({\omega}t) + b^2cos''({\omega}t) = cos({\omega}t)

Using the double angle formula for cosine, we can simplify this to:

-a^2cos({\omega}t) + b^2(-cos({\omega}t)) = cos({\omega}t)

Now, let's rearrange this equation to solve for x''(t):

x''(t) = (-a^2+b^2)cos({\omega}t)

We can see that this is a second order differential equation with a forcing function of cos({\omega}t). This means that the output of this equation will be a Fourier series with harmonics of cos({\omega}t), which is what you are looking for.

To make the output "warm", we can adjust the constants a and b. For example, if we choose a = 1 and b = 2, we will get an output with a stronger second harmonic, making it sound warmer.

I hope this helps you in your project. Good luck!

## 1. What is a nonlinear ODE?

A nonlinear ordinary differential equation (ODE) is a mathematical equation that describes the relationship between a function and its derivatives, where the function and its derivatives are not proportional to each other. In other words, the derivative of the function is not a constant multiple of the function itself.

## 2. What is harmonic output?

Harmonic output refers to a type of periodic motion or signal where the frequency of the motion or signal is an integer multiple of a fundamental frequency. In other words, the motion or signal repeats itself at regular intervals, with each repetition being a multiple of the fundamental frequency.

## 3. How is a nonlinear ODE with harmonic output different from a linear ODE?

A nonlinear ODE with harmonic output differs from a linear ODE in that the relationship between the function and its derivatives is not linear. This means that the function and its derivatives are not proportional to each other, making the solution to the ODE more complex and often involving techniques such as numerical methods or series solutions.

## 4. What is the importance of studying nonlinear ODEs with harmonic output?

Nonlinear ODEs with harmonic output have many applications in various fields of science and engineering, such as in the study of chaotic systems, oscillating systems, and many other physical phenomena. Understanding and solving these types of equations can provide valuable insights and predictions in these areas.

## 5. How are nonlinear ODEs with harmonic output solved?

Nonlinear ODEs with harmonic output can be solved using a variety of techniques, including numerical methods, series solutions, and approximate methods. The specific method used will depend on the complexity of the equation and the desired level of accuracy in the solution.

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