Transcendental Equations for Maple

[G01]

I am trying to use Maple to graph one variable vs. another. This is normally a simple issue, but the equations I have involving both of the variables is transcendental and it is impossible to solve for the variable I wish to be the independent variable. Does anyone know how I can go about graphing the one variable vs. the other in Maple? The equation has the following form:

$$y\tan(yx)=\sqrt{1-y}$$

Is there anyway I can graph y vs. a and not loose any solutions or is what I'm asking for impossible?

 

[Astronuc]

Can one simply use

$$x\,=\,\frac{1}{y}\,tan^{-1}\frac{\sqrt{1-y}}{y}$$

Bear in mind that the square root maybe negative as well as positive, but perhaps the negative is not physical and need not be considered.

 

[George Jones]

I am trying to use Maple to graph one variable vs. another. This is normally a simple issue, but the equations I have involving both of the variables is transcendental and it is impossible to solve for the variable I wish to be the independent variable. Does anyone know how I can go about graphing the one variable vs. the other in Maple? The equation has the following form:

$$y\tan(yx)=\sqrt{1-y}$$

Is there anyway I can graph y vs. a and not loose any solutions or is what I'm asking for impossible?

Maple, Mathematica, and, I think, Matlab all have commands for this. I only have access to Maple at home.

with(plots):
implicitplot(y*tan(y*x) = sqrt(1 - y), x=-3..3, y=1..-5, grid=[500,500]);

The graph is quite funky. This implicit function is very sensitive to numerical error, so you have to make sure what is plotted is meaningful. Play with the graph by making the grid finer, and by restricting the region. A plot with a finer grid may 10 minutes or even 30 minutes to produce, but this will be the only way to make the results are OK.

Can one simply use ...

This only produces a small part of the graph. tan defined on the whole real axis is a perfectly fine function, but it is not an invertible function. This is the tan function that G01 needs. tan^-1 is defined for a tan function whose domain is restricted to -pi/2 < x < pi/2

 

[G01]

Thanks George. That seemed to work. I have one more question. Maybe you can help. Is there anyway I can solve this equation and get multiple numerical answers? I tried using the "fsolve" command, but it will only give one solution in the range specified. Is there any command that can give me more than one numerical solution for a given range?

 

[George Jones]

Thanks George. That seemed to work. I have one more question. Maybe you can help. Is there anyway I can solve this equation and get multiple numerical answers?

I'm not sure what you mean. Do you mean for a particular fixed x?

 

[G01]

I'm not sure what you mean. Do you mean for a particular fixed x?

Yes, for a particular, fixed x.

Say I set x to 10. I can then use "fsolve" to find me a value for y that solves the equation. I can give the command a range in which to search for the y value and if it finds one it will return it. The problem is that I want to specify a range in which there is more than one possible solution, but "fsolve" will only return one of them. Is there a command, similar to "fsolve," that can return multiple solutions?