Solving $\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$ (using wolfram/mathematica)

Question

I would like to solve the following non-linear ordinary differential equation:

$$\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$$

-I need an analytic solution.

-I don't know where to start (the 2 terms make it difficult), therefore I tried to use Wolfram-Alpha/ Mathematica. See here

-It seems however that even Wolfram-Alpha / Mathematica is not able to solve it (computation time out).

-Can someone help me out?

Answer 1

Let $u=t-x$ ,

Then $x=t-u$

$\dfrac{dx}{dt}=1-\dfrac{du}{dt}$

$\therefore1-\dfrac{du}{dt}=\dfrac{A(1-t+u)}{t-t^2}-\dfrac{B(t-u)-C(t-u)^2}{(t-t^2)u}$

$u-u\dfrac{du}{dt}=\dfrac{Au}{t}+\dfrac{Au^2}{t-t^2}+\dfrac{Cu^2+(B-2Ct)u+Ct^2-Bt}{t-t^2}$

$u-u\dfrac{du}{dt}=\dfrac{Au}{t}+\dfrac{(A+C)u^2+(B-2Ct)u+Ct^2-Bt}{t-t^2}$

$u\dfrac{du}{dt}=\dfrac{(A+C)u^2}{t^2-t}+\left(\dfrac{B-2Ct}{t^2-t}-\dfrac{A}{t}+1\right)u+\dfrac{Ct^2-Bt}{t^2-t}$

This belongs to an Abel equation of the second kind.

In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.

Let $u=\dfrac{1}{v}$ ,

Then $\dfrac{du}{dt}=-\dfrac{1}{v^2}\dfrac{dv}{dt}$

$\therefore-\dfrac{1}{v^3}\dfrac{dv}{dt}=\dfrac{A+C}{(t^2-t)v^2}+\left(\dfrac{B-2Ct}{t^2-t}-\dfrac{A}{t}+1\right)\dfrac{1}{v}+\dfrac{Ct^2-Bt}{t^2-t}$

$\dfrac{dv}{dt}=-\dfrac{(Ct^2-Bt)v^3}{t^2-t}-\left(\dfrac{B-2Ct}{t^2-t}-\dfrac{A}{t}+1\right)v^2-\dfrac{(A+C)v}{t^2-t}$

Please follow the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2

Answer 2

I tried to use Mathematica to solve your problem and it solved pretty fast. Here is what I've done.

Answer 3

My solution:

ClearAll[A, B, Cc, x, t]

AsymptoticDSolveValue[ Derivative[1][x][t] == (A (1 - x[t]))/(t - t^2) - ( B x[t] - Cc x[t]^2)/((t - t^2) (t - x[t])), x[t], {t, A, 1}]

(* C1 + (-A + t) (-((1 - C1)/(-1 + A)) - ((-1 + 2 A) (1 - C1))/(-1 + A)^2 - (A (B C1 - Cc C1^2))/((A - A^2) (A - C 1)^2) + ((-1 + 2 A) (B C1 - Cc C1^2))/((-1 + A)^2 A (A - C1)) - (-B C1 + Cc C1^2)/((-1 + A) A (A - C1)) - A (-(((-1 + 2 A) (1 - C1))/((-1 + A)^2 A)) - ( B C1 - Cc C1^2)/((A - A^2) (A - C1)^2) + ((-1 + 2 A) (B C1 - Cc C1^2))/((-1 + A)^2 A^2 (A - C1)))) *)

Mind the x0 is set be selection to A without any restriction. The order of the asymptotic expansion is set to the lowest nonzero one.

AsymptoticDSolveValue[
 Derivative[1][x][t] == (A (1 - x[t]))/(t - t^2) - (
   B x[t] - Cc x[t]^2)/((t - t^2) (t - x[t])), x[t], {t, A, 2}]

---

C[1] + ((-A + t) (-A^2 + A C[1] + A^2 C[1] + B C[1] - A C[1]^2 - 
    Cc C[1]^2))/((-A + A^2) (A - C[1])) + 
 1/(2 (-A + A^2)^2 (A - C[1])^3) (-A + t)^2 (-A^4 + A^5 - A^3 B + 
    3 A^3 C[1] - 2 A^4 C[1] - A^5 C[1] + 3 A^2 B C[1] - A^3 B C[1] + 
    A B^2 C[1] + 2 A^3 Cc C[1] - 3 A^2 C[1]^2 + 3 A^4 C[1]^2 - 
    3 A B C[1]^2 + 2 A^2 B C[1]^2 - 5 A^2 Cc C[1]^2 - 
    3 A B Cc C[1]^2 + A C[1]^3 + 2 A^2 C[1]^3 - 3 A^3 C[1]^3 + 
    B C[1]^3 - A B C[1]^3 + 4 A Cc C[1]^3 + B Cc C[1]^3 + 
    2 A Cc^2 C[1]^3 - A C[1]^4 + A^2 C[1]^4 - Cc C[1]^4 - Cc^2 C[1]^4)

AsymptoticDSolveValue[
 Derivative[1][x][t] == (A (1 - x[t]))/(t - t^2) - (
   B x[t] - Cc x[t]^2)/((t - t^2) (t - x[t])), x[t], {t, A, 3}]
___
C[1] + ((-A + t) (-A^2 + A C[1] + A^2 C[1] + B C[1] - A C[1]^2 - 
    Cc C[1]^2))/((-A + A^2) (A - C[1])) + 
 1/(2 (-A + A^2)^2 (A - C[1])^3) (-A + t)^2 (-A^4 + A^5 - A^3 B + 
    3 A^3 C[1] - 2 A^4 C[1] - A^5 C[1] + 3 A^2 B C[1] - A^3 B C[1] + 
    A B^2 C[1] + 2 A^3 Cc C[1] - 3 A^2 C[1]^2 + 3 A^4 C[1]^2 - 
    3 A B C[1]^2 + 2 A^2 B C[1]^2 - 5 A^2 Cc C[1]^2 - 
    3 A B Cc C[1]^2 + A C[1]^3 + 2 A^2 C[1]^3 - 3 A^3 C[1]^3 + 
    B C[1]^3 - A B C[1]^3 + 4 A Cc C[1]^3 + B Cc C[1]^3 + 
    2 A Cc^2 C[1]^3 - A C[1]^4 + A^2 C[1]^4 - Cc C[1]^4 - 
    Cc^2 C[1]^4) + 
 1/(6 (-A + A^2)^3 (A - C[1])^5) (-A + t)^3 (-2 A^6 + 3 A^7 - A^8 - 
    3 A^5 B + 6 A^6 B - A^4 B^2 - 2 A^6 Cc + 10 A^5 C[1] - 
    13 A^6 C[1] + 2 A^7 C[1] + A^8 C[1] + 13 A^4 B C[1] - 
    25 A^5 B C[1] + 3 A^3 B^2 C[1] - 6 A^4 B^2 C[1] + A^2 B^3 C[1] + 
    14 A^5 Cc C[1] - 8 A^6 Cc C[1] + 8 A^4 B Cc C[1] - 
    20 A^4 C[1]^2 + 20 A^5 C[1]^2 + 5 A^6 C[1]^2 - 5 A^7 C[1]^2 - 
    23 A^3 B C[1]^2 + 42 A^4 B C[1]^2 - A^5 B C[1]^2 - 
    3 A^2 B^2 C[1]^2 + 12 A^3 B^2 C[1]^2 + 2 A B^3 C[1]^2 - 
    37 A^4 Cc C[1]^2 + 30 A^5 Cc C[1]^2 + 4 A^6 Cc C[1]^2 - 
    24 A^3 B Cc C[1]^2 + 13 A^4 B Cc C[1]^2 - 7 A^2 B^2 Cc C[1]^2 - 
    8 A^4 Cc^2 C[1]^2 + 20 A^3 C[1]^3 - 10 A^4 C[1]^3 - 
    20 A^5 C[1]^3 + 10 A^6 C[1]^3 + 21 A^2 B C[1]^3 - 
    36 A^3 B C[1]^3 + 3 A^4 B C[1]^3 + A B^2 C[1]^3 - 
    6 A^2 B^2 C[1]^3 + 49 A^3 Cc C[1]^3 - 42 A^4 Cc C[1]^3 - 
    15 A^5 Cc C[1]^3 + 27 A^2 B Cc C[1]^3 - 30 A^3 B Cc C[1]^3 - 
    2 A B^2 Cc C[1]^3 + 24 A^3 Cc^2 C[1]^3 - 6 A^4 Cc^2 C[1]^3 + 
    12 A^2 B Cc^2 C[1]^3 - 10 A^2 C[1]^4 - 5 A^3 C[1]^4 + 
    25 A^4 C[1]^4 - 10 A^5 C[1]^4 - 10 A B C[1]^4 + 16 A^2 B C[1]^4 - 
    3 A^3 B C[1]^4 - 35 A^2 Cc C[1]^4 + 26 A^3 Cc C[1]^4 + 
    21 A^4 Cc C[1]^4 - 14 A B Cc C[1]^4 + 21 A^2 B Cc C[1]^4 - 
    27 A^2 Cc^2 C[1]^4 + 15 A^3 Cc^2 C[1]^4 - 4 A B Cc^2 C[1]^4 - 
    6 A^2 Cc^3 C[1]^4 + 2 A C[1]^5 + 7 A^2 C[1]^5 - 14 A^3 C[1]^5 + 
    5 A^4 C[1]^5 + 2 B C[1]^5 - 3 A B C[1]^5 + A^2 B C[1]^5 + 
    13 A Cc C[1]^5 - 6 A^2 Cc C[1]^5 - 13 A^3 Cc C[1]^5 + 
    3 B Cc C[1]^5 - 4 A B Cc C[1]^5 + 14 A Cc^2 C[1]^5 - 
    12 A^2 Cc^2 C[1]^5 + B Cc^2 C[1]^5 + 4 A Cc^3 C[1]^5 - 
    2 A C[1]^6 + 3 A^2 C[1]^6 - A^3 C[1]^6 - 2 Cc C[1]^6 + 
    3 A^2 Cc C[1]^6 - 3 Cc^2 C[1]^6 + 3 A Cc^2 C[1]^6 - Cc^3 C[1]^6

)

I present this as:

Abel ordinary differential equation of the second kind a[t_]: = (t - t^2) t b[t_]: = -(t - t^2) f[t_]: = A t g[t_]: = A + B + A t h[t_]: = A + C k[t_]: = 0

k has to be unequal zero to identify this. This is in contrast to the Maple result which might be more flexible than Mathematica DSolve Ordinary Differential Equations. But if that would be solvable in Mathematica it should work for everybody and not for one.

DEtoolsodeadvisor; gives [_rational, [_Abel, 2nd type, class B]]

There is some inconsistency with Class B if one looks at odeadvisor%2FAbel2A: there is no Class B in the Maple online documentation for Abel ordinary differential equations.