Solve non-separateable equation

I have a equation like this . Its just a simplified form of my actual equation to illustrate my query. The output y and input time t are non seperatable.

I wanted to solve it in Matlab to find the value of y for the time interval t

I saw examples of fsolve and fzero. But all of them are for seperatable functions. How can I solve this one?

I tried this too. But Error

y^2 = @(x)y*x^2 +(1+x)/(y+1);
t = linspace(-3.5,3.5);
plot(t,y(t),t,zeros(size(t)),'k-')
xlabel('x')
ylabel('y(x)')

Answer 1

[John D'Errico]

The usual name for this is an implicit function. There is no relationship where you can solve directly for y as a function of x. Actually, in the function you wrote, it is technically possible, but you said this is not your real problem.)

Of course, this syntax is meaningless in MATLAB:

y^2 = @(x)y*x^2 +(1+x)/(y+1);

You have an implicit function. The simplest way to solve the problem is to use fimplicit to plot it.

Fxy = @(x,y) -y.^2 + y.*x.^2 + (1 + x)./(1 + y);
fimplicit(Fxy,[-3.5,3.5])
xlabel X
ylabel Y

As you can see in the plot, for ANY given x, there are multiple values of y. So it is not a single valued function, but one with multiple branches, and a singularity.