Unable to find explicit solution

Matlab says "Unable to find explicit solution" when trying to solve following non linear equation

-log(y) = psi(1/2+ x/y) - psi(1/2)

Here psi is digamma function

What is wrong with my code, below?

I am using matlab version R2021a

syms x y 
eqn = psi(1/2 + x/y)- psi(1/2)+ log(y)==0;
soly = solve(eqn,y)

Warning:
Unable to find explicit solution. For options, see help. 
> In sym/solve (line 317)
soly =
 
Empty sym: 0-by-1

[Matt J]

It seems very plausible to me that an explicit solution does not exist. Do you have reason to think otherwise?

[Tiku]

Thank you for the reply.

I see explicit solution doesn't exist.

That means in order to plot y vs x I should assigned the value of x and solve for y right?

Answer 1

[Matt J]

You can use fimplicit.

[Tiku]

Alright and thank you

[Matt J]

You're welcome, but if fimplicit does what you need, please Accept-click this answer.

[Tiku]

Actually I am interested in a plot of y versus x .

Like I said earlier, I took 6 x points and used them to find corresponding y solutions using vpasolve command but it turns out that plot didn't match with the one in the paper. The paper says they obtained plot from solving equation self consistently.

I used following code

syms x y nw;

nw=6;

x=[0.00 0.12 0.14 0.16 0.18 0.20];

gv=zeros(1,nw);

for i=1:nw

[gv(i)] = vpasolve(psi(1/2 + x(i)/y)- psi(1/2)+ log(y)==0,y);

end;

gv

plot(x,gv)

Could you please let me know where is my error?

[Matt J]

If the idea is to generate a plot, I don't know why aren't using fimplicit like I suggested. It generates a reasonable looking plot for me quite readily:

f=@(x,y) psi(1/2+ x./y) - psi(1/2) + log(y);

fimplicit(f)
xlabel 'x', ylabel 'y'
axis([0,0.14,0, 1]);

[Tiku]

Thank you for your constant help and now I used fimplicit like you suggested.

Running of following code gave me a plot which didn't match with the attached plot the one from the paper. I am trying to get first plot and interested in Ts and Tc for now.

f=@(x,y) psi(1/2+2.2*x./y) - psi(1/2) + log(y/3);
fimplicit(f)
figure(1);
%xlabel 'x', ylabel 'y'
axis([0,0.18,0, 3]);
hold on
g=@(x,y) psi(1/2+ x./y) - psi(1/2) + log(y);
fimplicit(g)
figure(2);
%xlabel 'x', ylabel 'y'
axis([0,0.14,0, 1]);
hold on

On varying x from 0 to 0.14 there should be only one phase then mixed phase for certain x and other phase after increasing x.So how can someone generate the plot like the first figure?

[Matt J]

Here's what I get.

f=@(x,y) psi(1/2+2.2*x./y) - psi(1/2) + log(y/3);
g=@(x,y) psi(1/2+ x./y) - psi(1/2) + log(y);

fimplicit(f)
hold on
fimplicit(g)
hold on

axis([0.11,0.19,0, 2]);

[Tiku]

Thank you for your help but I am still wondering how to get like the plot in the paper?

My apology in the first equation 0.2 value was missing which makes full code now

f=@(x,y) psi(1/2+2.2*x./y) - psi(1/2) + log(y/3);
g=@(x,y) psi(1/2+0.2*x./y) - psi(1/2) + log(y);
fimplicit(f)
hold on
fimplicit(g)
hold on
axis([0.11,0.19,0, 2]);

The above code gave two intersecting curves but still both phases are apppearing in left-right and up-down of the plot which doesn't match with the paper.

[Matt J]

We've no reason to trust the paper more than Matlab...

[Tiku]

I totally agree Matlab calulation is right. Actually after the plot I have to use some physics to ignore certain parts of the plot so at this stage both Matlab and paper is right.

So I really apppreciate your help and thank you.