Can anbody solve this simple equation?

Hello everybody.I have attached the equation 1 and equation 2 in picture,How the values on right hand side are obtained, can anybody write it step by step.I will be very thankful.Please ignore the the X and 0 in Rg(0) anf Rg(X)

[Walter Roberson]

How can they be obtained given what inputs?

Answer 1

[Stephan]

Hi,

to obtain values from your function use this:

fun_rg = @(R_g0,sigma_1,sigma_2,x)exp((x.^2.*(R_g0.^2-1.0).^2.*(1.0./2.0))./(R_g0.^2.*sigma_1.^2-R_g0.^2.*sigma_2.^2)).*sqrt(R_g0.^2)

This function handle takes values for: Rg(0), σ1, σ2 and x and gives back the value for Rg(x). For example with:

Rg(0) = 1.25
σ1 = 10
σ2 = 25
x = 3

>> fun_rg(1.25,10,25,3)

ans =

    1.2478

The code to get this function is:

syms sigma_b sigma_1 sigma_2 R_g(x) R_g0
a = sqrt((sigma_b^2 + sigma_2^2) / (sigma_b^2 + sigma_1^2))
b = (sigma_2^2 - sigma_1^2) * x^2
c = 2 * (sigma_b^2 + sigma_2^2) * (sigma_b^2 + sigma_1^2)
fun_sigma_b = sqrt((sigma_2^2 - R_g0^2 * sigma_1^2) / (R_g0^2 -1))
fun_Rg_x = R_g(x) == a * exp(-b/c)
fun_Rg_x = simplifyFraction(subs(fun_Rg_x,sigma_b, fun_sigma_b))
fun_rg = matlabFunction(rhs(fun_Rg_x))

where a,b,c are stepwise written parts of the equation for Rg(x), so that Rg(x) can be written in the form:

R_g(x) = a * exp(-b/c)

This step is not neede but helps seeing what happens. To see the results in a more mathematical notation have a look at the file attached.

EDIT :

To get the result from equation 2 starting from equation 1 use:

syms sigma_b sigma_1 sigma_2 R_g(x) R_g0
a = sqrt((sigma_b^2 + sigma_2^2) / (sigma_b^2 + sigma_1^2))
b = (sigma_2^2 - sigma_1^2) * x^2
c = 2 * (sigma_b^2 + sigma_2^2) * (sigma_b^2 + sigma_1^2)
fun_Rg_x = R_g(x) == a * exp(-b/c)
fun_Rg_x = subs(fun_Rg_x,x,0)
fun_Rg_x = simplify(expand(isolate(fun_Rg_x,sigma_b)))

This will give the equation as you wanted. The mathematical notation is attached as a .pdf-file.

Best regards

Stephan

[Haseeb Hassan]

Sir first of all appreciated efforts.This will be helpful in my future work, but i mean can you solve the equation 1 in picture 1 by mathematically.I mean how to simplify algebraically the equation 1 and to get the equation 2 in picture 2. I just need step by step mathematical solution of the equation 1.

[Stephan]

See my edited answer and the new attachment.

[Haseeb Hassan]

Sir Stephan Jung...You won the show.Thumbs up for such a nice and clear answer.