Trying to solve a rather difficult differential equation

[RYANDTRAVERS]

Homework Statement

Consider a system composed of two species X and Y with fractional populations x and y, respectively, where x+y=1. The two species interact in such a way that the differential equation for x is:

\begin{equation}
\frac{dx}{dt}=xyA_{0}e^{-\alpha t}
\end{equation}

where $A_{0}$ and $\alpha$ are non-negative constants. Solve the equation by separation of variables and hence show that the solution for x(0) = $x_{0}$ is:

Photo attached- too long to write out!

2. The attempt at a solution

Again... attached. The problem that I am having is that I can't make x the subject of the equation because I end up with x/(x-1) on the left hand side.

 

[Chestermiller]

Are you saying that you don't know how to solve that final algebraic equation for x?

 

[SammyS]

Homework Statement

Consider a system composed of two species X and Y with fractional populations x and y, respectively, where x+y=1. The two species interact in such a way that the differential equation for x is:

\begin{equation}
\frac{dx}{dt}=xyA_{0}e^{-\alpha t}
\end{equation}

where $A_{0}$ and $\alpha$ are non-negative constants. Solve the equation by separation of variables and hence show that the solution for x(0) = $x_{0}$ is:

Photo attached- too long to write out!

2. The attempt at a solution

Again... attached. The problem that I am having is that I can't make x the subject of the equation because I end up with x/(x-1) on the left hand side.

Those sideways images are very difficult to read.

I did use the 'Windows' snipping tool to show the solution you are to verify, then pasted it into a word processor app. & rotated it.

Chet's got the rest.

 

[RYANDTRAVERS]

Don't worry, I was a little tired last night doing a 5 hour practice paper. I've got it now... silly me.

 

[SammyS]

Good.

To make x the subject either of the following forms might have helped.

$\displaystyle\ \frac{x}{x-1}=\frac{1}{1-1/x}=1+\frac{1}{x-1}\$