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I meet a first-order ODE

$$\frac{dy}{dt}=\frac{a(\ln\frac{1-c}{1-y})^3}{\frac{b-y}{1-y}+\ln\frac{1-c}{1-y}},$$ where $a,b,c$ are constants. The ODE is subjected to the initial condition $y(t=0)=y_0$. I simply use

DSolve[{y'[t] == 
a (Log[(1 - c)/(1 - y[t])])^3/((b - y[t])/(1 - y[t]) + 
 Log[(1 - c)/(1 - y[t])]), y[0] == y0}, y[t], t]

to seek for the solution. It gives me a InverseFunction of a long expression including an exponential integral function $Ei$. I know there is few possible to get an explicit expression for y(t). So I am not trying to do it.

My questions are:

Is there any possible to assign a special method for DSolve to obtain an analytical solution, say, separation of variables.

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    $\begingroup$ See Slot for the first question. $\endgroup$
    – Sungmin
    Oct 22, 2015 at 7:37

1 Answer 1

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You can tell Mma that. Try this:

    Integrate[
 1/(a (Log[(1 - c)/(1 - y)])^3/((b - y)/(1 - y) + 
       Log[(1 - c)/(1 - y)])), y]

(* (-b + y + (-1 + y) Log[(-1 + c)/(-1 + y)] + (-1 + 
    c) ExpIntegralEi[-Log[(-1 + c)/(-1 + y)]] Log[(-1 + c)/(-1 + 
    y)]^2)/(2 a Log[(-1 + c)/(-1 + y)]^2)
*)

which is equal to t+C, obviously. That is your solution "by parts". Another story is that you cannot then solve it in terms of y. But sometimes it is not necessary.

Have fun!

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