# How can I solve this equation?

Solve[1 - ((1 + x^α)^(-β θ) (1 + θ + β θ Log[1 + x^α]))/(1 + θ) == p, x]


Solve cannot solve this equation. Note that β, θ, α and x are all positive but I don't know how to implement that in the Solve function.

• I am not sure if you can find an analytic result for $x$ but you can surely find a numerical result for $x$ by using FindRoot or NSolve. – optimal control Feb 25 '17 at 14:28
• @Ahmed Abo-Zaid can you please modify your question so it is clear what equation you are trying to solve. Otherwise your question may not help others in future – Ali Hashmi Feb 27 '17 at 18:12

It is best to see what happens on the Plot

α := 5;
θ := 5;
β := 5;
p := 1;
eq = 1 - ((1 + x^α)^(-β θ) (1 + θ + β*θ Log[1 + x^α]))/(1 + θ)-p

Plot[eq, {x, -1, 1}] Using FindInstance

sol1 = FindInstance[eq == 0, x, Reals, 5]
(*{{x -> -((-1 + E^(6/25))^(1/5)/E^(6/125))}}*)
sol2 = FindInstance[eq == 0, x, Complexes, 5]
(*{{x -> (-1 + 1/E^(6/25))^(
1/5)}, {x -> -(-1)^(1/5) (-1 + 1/E^(6/25))^(1/5)}, {x -> (-1)^(
2/5) (-1 + 1/E^(6/25))^(
1/5)}, {x -> -(-1)^(3/5) (-1 + 1/E^(6/25))^(1/5)}, {x -> (-1)^(
4/5) (-1 + 1/E^(6/25))^(1/5)}}*)


FindRoot Needs a good starting point.

 FindRoot[eq, {x, -0.8}]
(*{x -> -0.734222}*)


NSolve It is better, but in some cases.

NSolve[eq == 0, x]// TableForm
(*{
{x -> -0.734222},
{x -> -0.226887 - 0.698287 I},
{x -> -0.226887 + 0.698287 I},
{x -> 0.593998 - 0.431565 I},
{x -> 0.593998 + 0.431565 I}
}*)


You can also use:

 NSolve[{eq == 0, -2 < x < 2}, x]
{{x -> -0.734539}}
(*Only find reals solution*)
NSolve[{eq == 0, -1 < x < 0}, x]
NSolve[{eq == 0, -2 < x < 0}, x]


Symbolic solution.

Using powerfull Maple, it seems can solve :P where:

$\{\text{$\_$Z7}\in \mathbb{Z},\text{$\_$Z8}\in \mathbb{Z}\}$

with: z7=0 only exists solutions.

 z7 = 0;
x = Exp[(Log[(Exp[(-ProductLog[z7, (1 + θ)*(p - 1)*Exp[-1 - θ]] - 1 - θ)/(β*θ)] - 1)] + 2*I*Pi*z8)/α]


$$x=\exp \left(\frac{\log \left(\exp \left(\frac{-W_{\text{z7}}((1+\theta ) (p-1) \exp (-1-\theta ))-1-\theta }{\beta \theta }\right)-1\right)+2 i \pi \text{z8}}{\alpha }\right)$$

 Table[x // N, {z8, 0, 5}] // Chop // TableForm
(*{
{0.593998 + 0.431565 I},
{-0.226887 + 0.698287 I},
{-0.734222},
{-0.226887 - 0.698287 I},
{0.593998 - 0.431565 I},
{0.593998 + 0.431565 I}
}*)

• thanks, it seems that no escape from numerical solutions. – Ahmed Abo-Zaid Feb 26 '17 at 19:11
• @AhmedAbo-Zaid . Last equation is symbolic solution.Remove //N and //Chop symbols from Table. – Mariusz Iwaniuk Feb 26 '17 at 21:34

A numerical solution is possible, I gave some arbitrary values to parameters that you can change ;

FindRoot[1 - ((1 + x^\[Alpha])^(-\[Beta] \[Theta]) (1 + \[Theta] + \[Beta] \
\[Theta] Log[1 + x^\[Alpha]]))/(1 + \[Theta]) == p /. \[Alpha] -> 5 /. \[Beta]-> 5 /. \[Theta] -> 5 /. p -> 1, {x, 0.5}]


It gives

{x -> 2.30848}