Unable to solve a set of equations with Solve

Question

I'm trying to solve the following equation using Mathematica 9.0.1.0 :

Solve[E^(-σ τ) k (1 + T σ)^-n == Subscript[M, 0] && 
  2*E^(-σ τ) k (1 + T σ)^(-1 - n) (n T + τ + T σ τ) ==
    Subscript[M, 1] && 
  1/2 E^(-σ τ) k (1 + T σ)^(-2 - n) (n T^2 + n^2 T^2 + 2 n T τ + 
      2 n T^2 σ τ + τ^2 + 2 T σ τ^2 + 
      T^2 σ^2 τ^2)*6 == Subscript[M, 2] && 
  k (-(1/6) E^(-σ τ) n (1 + n) (2 + n) T^3 (1 + T σ)^(-3 - n) - 
      1/2 E^(-σ τ) n (1 + n) T^2 (1 + T σ)^(-2 - n) τ - 
      1/2 E^(-σ τ) n T (1 + T σ)^(-1 - n) τ^2 - 
      1/6 E^(-σ τ) (1 + T σ)^-n τ^3) == Subscript[M, 3], {k, T, 
  n, τ}]

Mathematica outputs the following error:

Solve::nsmet: This system cannot be solved with the methods available to Solve. >>

And when I try to use "SolveAlwyas" instead of "Solve" I get the following errors :

Solve::incnst: Inconsistent or redundant transcendental equation. After reduction, the bad equation is 1-((1+T σ)^n)^(1/n) == 0. >>
Solve::incnst: Inconsistent or redundant transcendental equation. After reduction, the bad equation is -1+((1+T σ)^n)^(1/n) == 0. >>
Solve::incnst: Inconsistent or redundant transcendental equation. After reduction, the bad equation is 1-((1+T σ)^(-1-n))^(1/(-1-n)) == 0. >>
General::stop: Further output of Solve::incnst will be suppressed during this calculation. >>

This system can be solved.

Is there any way to solve this equations with Mathematica without giving numerical values to the parameters ?

Answer 1

It can be solved in this way

From my observation, your are solving this system:

$$ g(\sigma) = e^{-\sigma \tau}k(1+T\tau)^{-n} $$

$$ g(\sigma)=A_0 $$ $$ g'(\sigma)=A_1 $$ $$ g''(\sigma)=A_2 $$ $$ g'''(\sigma)=A_3 $$

My constants $A$ differ from your $M$ by some factor, you should be able to see it.

First, eliminate $k$

Reduce[g == A0 && g1 == A1 && g2 == A2 && g3 == A3, k]

you will get,

(1 + T \[Sigma] != 0 && A0 n != 0 && A1 + A0 \[Tau] != 0 && 
    E^(\[Sigma] \[Tau]) (1 + T \[Sigma])^(1 + 2 n) != 0 && 
    A1 == (-A0 n T - A0 \[Tau] - A0 T \[Sigma] \[Tau])/(
     1 + T \[Sigma]) && 
    A2 == (A1^2 + A1^2 n + 2 A0 A1 \[Tau] + A0^2 \[Tau]^2)/(A0 n) && 
    A3 == (-A1^2 A2 + 2 A0 A2^2 - 2 A1^3 \[Tau] + 3 A0 A1 A2 \[Tau])/(
     A0 (A1 + A0 \[Tau])))) && 
 k == A0 E^(\[Sigma] \[Tau]) (1 + T \[Sigma])^n

And then solve for the variables without $k$

Solve[A1 == (-A0 n T - A0 \[Tau] - A0 T \[Sigma] \[Tau])/(
   1 + T \[Sigma]) && 
  A2 == (A1^2 + A1^2 n + 2 A0 A1 \[Tau] + A0^2 \[Tau]^2)/(A0 n) && 
  A3 == (-A1^2 A2 + 2 A0 A2^2 - 2 A1^3 \[Tau] + 3 A0 A1 A2 \[Tau])/(
   A0 (A1 + A0 \[Tau])), {T, \[Tau], n}]

By putting back the factors relating $A$ and $M$ , you will get the answer.