Unable to find explicit solutions to an equation

Hi,

I am trying to resolve the following equation :

x*atan(1/x)=y for x

My code is:

syms x y
eqn=(x*atan(1/x))==y
eqn=rewrite(eqn, 'log');
solx=solve(eqn, x, 'IgnoreAnalyticConstraints',1)
pretty(solx)

But I am getting the following error :

Any ideas?

Thank you in advance.

Answer 1

[Walter Roberson]

There is no closed form formula for that.

[Walter Roberson]

y_values = sort(rand(1,10))

y_values = 1×10

0.0586 0.2845 0.3234 0.3530 0.3767 0.4559 0.5491 0.5873 0.5990 0.7952

opts = optimset('display', 'none');
x_values = arrayfun(@(Y) fsolve(@(x) x*atan(x) - Y, 0.75, opts), y_values);
X = linspace(0,1);
Y = X.*atan(X);
plot(X, Y, '-', x_values, y_values, 'r+')

For the red +, in each case the x coordinate is the calculated one for that particular y, and you can see that it matches up perfectly with the forward calculation of x*atan(x) shown by the line.

[Plamen Bonev]

Thank you so much Walter :)

Answer 2

[Image Analyst]

I'm not sure what you want to "resolve" or solve. This is what I see when I plot it:

fontSize = 20;
x = linspace(-2*pi, 2*pi, 1000);
y = x .* atan(1./x);
plot(x, y, 'LineWidth', 2);
grid on;
xlabel('x', 'FontSize', fontSize);
ylabel('y', 'FontSize', fontSize);
title('y = x .* atan(1./x)', 'FontSize', fontSize);

Now, explain exactly what is supposed to be equal to what.

[Plamen Bonev]

Thank you for you answer.

Actually my goal is :

I have experimental data on the y-values and I want to find x corresponding to the y value by solving the equation for x...

[Walter Roberson]

I think the point is, given a y, find the x such that y == x * atan(1/x)

It looks to me as if outside of y in [0 1] that the solution would have to be complex valued.

[Plamen Bonev]

Yes exactly Walter.

Do you have any idea on how to solve it then?

Thank you for your help.

[Image Analyst]

I was about to give you a numerical solution for when y is between 0 and 1. Do you ever have that case? It sounds like, from your comment to Walter, that all your y values are below 0 or above 1, so I didn't program up a solution. Let me know if you need one, and to what accuracy (how many decimal places) you require.

[Plamen Bonev]

Hello,

My y values are neither below 0 nore above 1. Actually, they are mostly between 0.5 and 1.

Could you give me a solution in this range.

I will really appreciate your help.

Thank you.

[Image Analyst]

Oh, at first you asked Walter for a solution for values outside the [0,1] range. Anyway, it looks like Walter gave you a solution for the [0,1] range and you've accepted that so I guess we're done. It looks like you actually wanted x*atan(x), like in his solution, rather than x*atan(1/x) like you originally asked for. But it would be a trivial modification to his answer if you wanted 1/x instead of x as the atan argument.

[Walter Roberson]

ah, No wonder the graph didn't look like what I was expecting...

4 AM mistake.

[Plamen Bonev]

Thank you so much both for you answers.

Image Analyst, it is true that I did not pay attention, but you are right.

I did not tell that I had values outside the [0,1] range though.

It is as you say, a simple modification to the equation in order to account for the (1/x) and not for the (x) dependence.

Thank you so much.

Cheers.

[Plamen Bonev]

Hi again,

Actually the fsolve does not work with my licence (cause it requires the Optimization Toolbox).

Do you have any other idea without using fsolve?

Thank you again...

[Walter Roberson]

Substitute fzero()

[Plamen Bonev]

Thanks a lot Walter.