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

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In summary, an ODE (ordinary differential equation) is a mathematical equation used to describe the relationship between a function and its derivatives. It is used in science and engineering to model various phenomena. The notation $\frac{dx}{dt}$ represents the derivative of the function x with respect to the variable t in the ODE $\frac{dx}{dt}=ax(b-x)$, describing the rate of change of x over time. To solve an ODE, a function or set of functions that satisfy the equation must be found using methods such as separation of variables, substitution, or an integrating factor. The parameters a and b in the ODE $\frac{dx}{dt}=ax(b-x)$ represent the growth rate or decay rate of
  • #1
Acut
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How can this equation be solved?

[tex]\frac{dx}{dt}[/tex]=ax(b-x)
 
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  • #2
By separation of variables.

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

Now you can integrate both sides.
 
  • #3
The integral of the dx side requires decomposition into partial fractions.
 
  • #4
Many thanks!
I'm a bit rusty in solving ODE's and was having a hard time trying to solve this one..
 
  • #5
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.
 

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

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 used to model many physical and natural phenomena in science and engineering.

2. What does $\frac{dx}{dt}$ mean in the ODE $\frac{dx}{dt}=ax(b-x)$?

The notation $\frac{dx}{dt}$ represents the derivative of the function x with respect to the variable t. In this ODE, it is describing the rate of change of x over time.

3. How do you solve an ODE?

To solve an ODE, you need to find a function or set of functions that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using an integrating factor.

4. What is the significance of the parameters a and b in the ODE $\frac{dx}{dt}=ax(b-x)$?

The parameter a represents the growth rate or decay rate of the function x, while b represents the carrying capacity or maximum value that x can reach. These parameters can provide insight into the behavior of the solution to the ODE.

5. Can an ODE be solved analytically?

Not all ODEs can be solved analytically, meaning a closed-form solution cannot be obtained. In these cases, numerical methods are often used to approximate the solution. However, some ODEs do have analytical solutions, which can provide a deeper understanding of the system being modeled.

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