# Solving ODE: Need Help and Ideas

• tandoorichicken
In summary, the conversation discusses solving a specific ODE using different methods such as using dy/dt notation, integrating, and using partial fractions. They also mention using a hyperbolic tangent substitution. Ultimately, they all arrive at the same solution for y(t).
tandoorichicken
I need help in solving the following ODE.
$$y'(t) = \frac{k}{M}y(M-y)$$

Not quite sure what to do. I multiplied everything out so I was left without any parenthesis, but I don't know where to go from there. Any ideas/hints would be appreciated.

i think you might see it better with dy/dt notation
what you have is $$\frac{dy}{dt} = \frac{k}{M} y(M-y)$$
which will become
$$\frac{dy}{y(M-y)} = \frac{k}{M} dt$$
integrate away!
use partial fractions, seems to work just fine

Last edited:
Thanks, I think I got it.

i got this answer by teh way
$$y(t) = \frac{MCe^{kt}}{1+Ce^{kt}}$$
$$C = e^{C_{1}}$$ from the integration

It could be done without partial fractions,using a hyperbolic tangent substitution.

Daniel.

You really like those hyperbolics don't you

Thanks everyone. Yes I did get the same answer as stunner.

## 1. What is an ODE?

An ODE (Ordinary Differential Equation) is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model systems in science and engineering.

## 2. How do I solve an ODE?

To solve an ODE, you can use various methods such as separation of variables, substitution, or using numerical techniques like Euler's method or Runge-Kutta method. The method used depends on the type and complexity of the ODE.

## 3. What are some real-world applications of ODEs?

ODEs have various applications in fields such as physics, engineering, economics, and biology. For example, they can be used to model the motion of objects, describe the growth of populations, or analyze the behavior of chemical reactions.

## 4. What are the initial conditions in an ODE?

Initial conditions refer to the values of the dependent variable and its derivatives at the starting point of the ODE. These values are necessary to find a specific solution to the ODE.

## 5. How can I check if my solution to an ODE is correct?

There are several ways to check the correctness of your solution to an ODE. You can verify if it satisfies the original equation, check if it meets the initial conditions, and compare it to known solutions or numerical approximations. It is also helpful to double-check your work and use multiple methods to solve the ODE.

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