Numerical solution of equation with free parameters

I have an equation which is not analytically solvable,
thus I need to solve it numerically.
But, this equation has some free parameters which I want to replace them with some numbers after solving the equation

.
Let me ask a very simplified version of the question,
then I can solve the original one by the help of this simplified version.

I have an equation with the form of $$a\space x^5+b\space sin(x)+3=0$$.
Now, I want to plot the root of this equation in the a-b coordinate system.

In another word,
I want to fix the root and see how it behaves by changing our free parameters, $$a$$ and $$b$$.

You can try ContourPlot3D:

ContourPlot3D[a x^5 + b Sin[x] + 3 == 0,
{a, -1, 1}, {b, -1, 1}, {x, -2 Pi, 2 Pi},
AxesLabel->{"a","b","x"}]


Alternatively, use ContourPlot to show combinations of a and b that make a given x the root and vary x interactively using Manipulate:

Manipulate[ContourPlot[a x^5 + b Sin[x] + 3 == 0, {a, -1, 1}, {b, -1, 1},
FrameLabel -> {"a", "b"}],
{{x, Pi/2}, -2 Pi, 2 Pi, Appearance -> "Labeled"}]


• Thank you so much for your help. How can I transform this plot into two-dimensional a-b coordinate system for constant x? Jun 19 '19 at 0:31