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

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SUMMARY

The ordinary differential equation (ODE) $\frac{dx}{dt}=ax(b-x)$ can be solved using the method of separation of variables. By rearranging the equation to $\frac{dx}{ax(b-x)}=dt$, integration of both sides is required. The left-hand side necessitates partial fraction decomposition, specifically into the terms $\frac{1}{abx} - \frac{1}{ab(b-x)}$. This approach allows for the successful integration of the equation.

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How can this equation be solved?

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

<br /> \frac{dx}{ax(b-x)}=dt<br />

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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