Question

Find the explicit solution for:
dX/dt=(x-1)(2x-1), ln(2x-1/x-1)=t

Answer 1

`(\mathrm dx)/(\mathrm dt)=(x-1)(2x-1)`

is a separable ODE, as

`(\mathrm dx)/((x-1)(2x-1))=\mathrm dt`

Decompose the left side into partial fractions:

`\frac1{(x-1)(2x-1)}=\frac1{x-1}-\frac2{2x-1}`

Then integrating both sides gives

`\displaystyle\int\left(\frac1{x-1}-\frac2{2x-1}\right)\,\mathrm dt=\int\mathrm dt`

`\ln|x-1|-\ln|2x-1|=t+C`

Solve for
`x(t)`:

`\ln\left|(x-1)/(2x-1)\right|=t+C`

`(x-1)/(2x-1)=e^(t+C)=Ce^t`

`x-1=(2x-1)Ce^t`

`x(1-2Ce^t)=1-Ce^t`

`\implies\boxed{x(t)=(1-Ce^t)/(1-2Ce^t)}`