Please help me solve this Ln equation.really seems to be unsolvable?

Question

i know there is a way to solve this equation.but don't know the way :D
please help me.
the function is really simple [!!] :
x + x * ln(x) +a*x=2
solve it for x (we know the value of a)
ln=natural logarithm
thank u very much.

Answer 1

Let's factor the x:
f(x) = x*(1 + a + lnx) - 2 = 0
This equation cannot be solved analytically, but you can use an iterative approach to get an approximate solution:
One such approach is the fixed point iteration (pure calculator work). Write the equation as
x = 2 / (1 + a + lnx)
Let's consider the case where a = 2
x = 2/(3 + lnx)
Start with an initial guess of x = 1, then compute the RHS. Set that value to x and go again. After a few iterations, you end up with
x = 0.7408

NOTE that if a=1, x=1 is an exact solution. So that's why I chose a different value for a.
The only thing about the fixed point iteration is that it's not always stable. It needs to be written in the right form. Try it with the last line of Sciman's result and see:
x = e^[(2-ax-x)/x]

Answer 2

I assume the question is Solve for x:
x + xln(x) + ax = 2.
This might have 0 answers (i.e. no exact solution possible, or just no solution), one answer, or an infinte number of answers.

Anyway:
x + x*ln(x) +ax =2
x*ln(x) = 2-ax-x
ln(x) = (2-ax-x)/x

Now base to the power of log = number, and the base here is the constant e (approx 2.718)

So x = e^[(2-ax-x)/x], which perhaps looks worse, but in an exam one would get attempt marks.
Addendum:
I tried mathematica with this problem, and it couldn't handle it.

I let a = -2, -1, 0, 1, and a few more and solved these *approximately* using Deadline. Each equation had only one root, and only one root was exact, when x=1. I tried to approximate match the answers in Excel using various mathematical models, and plenty of guessing and nothing worked.

I went back to algebra methods by hand, e.g. substitute something for something..) but got no new success.

I didn't use infinite series because I am not good enough at the that.

My conclusion/ guess: For any given value of a.
1) There may exist a function which may or may not already have been discovered e.g. cosh(5), inverse cosh(5). This function might be used embedded in an expression or on its own.
2) As Godel showed around 1930, exact solutions may not be findable. Using the maths we now have unless we create new axioms.

Answer 3

x( 1 + a + ln(x) ) = 2
This is a nonlinear equation. Solve numerically by iteration, defing a sequence x(0), x(1),..., x(n) closer and closer to the solution.
For example, define
x(0) = 1
x(1) = 2 / ( 1 + a + ln(x(0)) )
= 2/ (1 + a)
...
suppose x(n) is defined, then
x(n+1) = 2 / ( 1 + a + ln(x(n)) )
you'll get closer and closer to the solution

Answer 4

Based on Sciman's comments, I agree that an infinite number of solutions is possible. Without knowing the value of a, it's impossible to speculate on x.

For example, if you set a = 0, x is approx 1.45, so just assigning a an arbitrarily small value, x would still be solvable at about 1.45

Answer 5

Yoy'll have to use numerical methods such as Newton-Raphson method to solve this equation. Also, you'll have to know the precise value of a.
If you really want to know, I'll attempt to solve it and send you an e-mail. I'll leave my e-mail open for 5 or 6 hours. If not, do not worry about replying me. I want the precise value of a. I'll give it a try.