Understanding the Role of Constants in First Order Differential Equations

In summary, the original problem was to solve dy/dx = x(1-x). The provided solution was y = (x^2)/2 - (x^3)/3 + C, but there were some discrepancies with the integration of dt and a possible typo with the derivative being dx/dt instead of dy/dx. After further discussion, it was concluded that the problem was most likely supposed to be dx/dt.
  • #1
converting1
65
0
solve dy/dx = x(1-x)

I got y = (x^2)/2 - (x^3)/3 + C

however in the solutions they've gotten:

259y0kp.png


where did t come from?
 
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  • #2
It came when they integrated dt on the right hand side, 2 steps above where you marked the solution.
 
  • #3
It looks to me like there is a typo in the problem. y disappeared! I think they meant the derivative to be dx/dt, not dy/dx. Your solution to dy/dx=x(1-x) is correct.
 
  • #4
Oh nice spot!
 
  • #5
Dick said:
It looks to me like there is a typo in the problem. y disappeared! I think they meant the derivative to be dx/dt, not dy/dx. Your solution to dy/dx=x(1-x) is correct.

Yeah it definitely looks like the problem was supposed to be dx/dt
 
  • #6
thanks guys
 

FAQ: Understanding the Role of Constants in First Order Differential Equations

What is a first order differential?

A first order differential is a type of mathematical equation that describes the relationship between a function and its derivative. It is written in the form dy/dx = f(x), where y is the dependent variable and x is the independent variable.

What is the difference between a first order differential and a second order differential?

The main difference between a first order differential and a second order differential is the number of derivatives involved. A first order differential involves only one derivative, while a second order differential involves two derivatives.

How are first order differentials used in science?

First order differentials are used in a variety of scientific fields, such as physics, biology, and engineering. They can be used to model and analyze many physical and biological processes, such as motion, growth, and decay.

What is the process for solving a first order differential equation?

The process for solving a first order differential equation involves isolating the dependent variable, integrating both sides of the equation, and then solving for the constant of integration. This process may also involve using mathematical techniques such as separation of variables or substitution.

Can first order differentials be used to predict future behavior?

Yes, first order differentials can be used to predict future behavior of a system or process. By solving the differential equation, we can determine the functional relationship between the dependent and independent variables, allowing us to make predictions about future behavior based on the current state of the system.

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