Double solutions and plotting transcendental equations

Question

I have the following transcendental equation:

$y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$

and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x \leq5$ interval) and $y>0$, for various values of $C$.

My computational approach was to calculate a numeric value for the R.H.S., using N number of increments in $x$'s interval and a pre-determined $C$, and then, for each of the values, solve for the corresponding value of y (using the Newton method, for e.g.)

However, in doing this (part-successfully), it would seem that I am neglecting a second value, a sort-of double root, as it were. For e.g.:

Say $C=0$ and $x=1$, giving: $y^2 - \log(y)^2 = 4$

I need to find two (or more?) positive solutions for y, and I'm really struggling. There's some serious self-doubt going on here and I'm just not seeing that second solution.

Big thanks in advance.

Answer 1

Why not just plot it as an implicit function? Maple, for example, has the implicitplot command.

plots[implicitplot](y^2 - log(y)^2 = 4*log(x) + 4/x + 1, x = 0 .. 5, y = 0 .. 5);

But as for your search for a second solution: $y^2 - \log(y)^2$ is increasing on $(0,\infty)$, so there is never more than one $y$ for a given $x$.