Solving ODE: $\frac{dx}{dt}=ax(b-x)$

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Discussion Overview

The discussion revolves around solving the ordinary differential equation (ODE) given by $\frac{dx}{dt}=ax(b-x)$. Participants explore methods for integrating this equation, focusing on the technique of separation of variables and the use of partial fractions.

Discussion Character

  • Technical explanation, Mathematical reasoning, Homework-related

Main Points Raised

  • One participant asks how to solve the ODE.
  • Another participant suggests using separation of variables as a method for solving the equation.
  • A third participant notes that the integral on the left side requires decomposition into partial fractions.
  • A later reply expresses gratitude and mentions difficulty in solving ODEs, indicating a personal context for the discussion.
  • Another participant provides a specific approach to partial fraction decomposition for the integral, breaking it into two terms for integration.

Areas of Agreement / Disagreement

Participants generally agree on the method of separation of variables and the need for partial fraction decomposition, but there is no consensus on the final solution or any specific outcomes from the integration process.

Contextual Notes

Some participants express uncertainty about the integration process and mention being rusty in solving ODEs, indicating potential gaps in understanding or experience with the topic.

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How can this equation be solved?

[tex]\frac{dx}{dt}[/tex]=ax(b-x)
 
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By separation of variables.

[tex] \frac{dx}{ax(b-x)}=dt[/tex]

Now you can integrate both sides.
 
The integral of the dx side requires decomposition into partial fractions.
 
Many thanks!
I'm a bit rusty in solving ODE's and was having a hard time trying to solve this one..
 
dx/(ax(b-x)) = dx/abx + dx/ab(b-x)
= dx/abx - d(b-x)/ab(b-x)
and then you can integrate these terms.
 

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