Various DE problems and related stuff - help

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In summary, the conversation discusses the struggles of a student in a DE class, where the focus is on solving equations. The teacher has assigned a lengthy homework sheet, causing confusion for the student. They are attempting to start with the first problem, which involves drawing a direction field and using variables such as x, t, r, b, and a. The student is unsure of how to approach the problem and asks if they can arbitrarily set some constants to 1. The other person confirms that a, b, and r are constants, but advises against setting any constant to 0 as it may affect the equation's generality.
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WirelesssTouc
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My DE class has been 100% about solving equations giving an equation. But my teacher seems to think giving us a very wordy sheet of homework would be fun and I just seem to be very lost. Trying to go through this systematically starting with the first problem and I've no idea what to do.

I have to draw a direction field. I figured I could plug in various values of x and t to just get dx/dt, but I don't know what to do about the other variables, r, b, and a. Can I simply set r=1, a=1, and b=0, arbitrarily, since they would be random constants in the logistics equation in the first place? So lost.

Attached is all the assignment.
 

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Yes, a, b, and r are constants. I would be a little careful about setting any constant to 0. That may remove important parts of the equation and so not be as general as you would like to.
 

1. What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes the relationship between a function and its derivatives, and is commonly used in mathematical models to describe natural phenomena or physical systems.

2. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve a single independent variable, while PDEs involve multiple independent variables. SDEs incorporate randomness into the equation.

3. What are some real-world applications of differential equations?

Differential equations have many real-world applications, such as in physics, engineering, economics, and biology. They can be used to model the motion of objects, the spread of diseases, the growth of populations, and the behavior of financial markets, among others.

4. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some ODEs can be solved analytically using integration or other mathematical techniques, while others may require numerical methods such as Euler's method or Runge-Kutta methods. PDEs and SDEs often require more advanced numerical methods for solving.

5. How can differential equations be used in data analysis?

Differential equations can be used in data analysis to model and predict the behavior of complex systems. By fitting data to a differential equation model, we can gain insight into the underlying dynamics of the system and make predictions about future behavior. This is commonly used in fields such as economics, epidemiology, and ecology.

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