# Questions tagged [implicit-function]

This tag is for questions relating to "implicit function", a function or relation in which the dependent variable is not isolated on one side of the equation.

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### Reference Request: Implicit Difference Equations

I know that there are some studies on implicit differential equations such as $$f(x, y, y') = 0.$$ I did some search but found very few results on the discrete version---implicit difference ...
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### What are the conditions for a function to be a unique implicit function?

My question is pertaining to part b. What are the conditions for a function to be a unique implicit function? Do we only have to check if the partial derivative at (x0, y0) evaluate to a number other ...
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### How can I show that certain implicit functions can partition $\mathbb R^2$?

Consider the implicit function defined by the equation $$f(x,y) = 0 \tag{1} \label{eq1}$$ where $f : \mathbb R^2 \rightarrow \mathbb R$ is some continuous function. Suppose that the implicit ...
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### Asymptotic behaviour of implicit functions

Suppose we have an implicit equation $F\left(x,y\right)=0$ which we know defines $y = y(x)$ as a function of $x$. Are there sufficient or necessary conditions under which we can obtain information ...
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### Topology of sets defined by real-valued functions

Suppose I have a topological space $S$ and a continuous real-valued function $f:S \to \mathbb R$. I can define sets like: \begin{align} A &= \{x \in S : f(x) = 0 \} \\ B &= \{x \in S : f(x) \...
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### Can 'f' be called a function in the given problem?

I know that if a variable $z = f(x,y)$, then $z$ or $f$ is a function of $x$ and $y$. Consider $f = xy^2+y=5.$ Clearly, $xy^2+y=5$ is a curve on the x-y plane. $y$ and $x$ are implicitly related, and ...
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### Express y in terms of x for equation $y = 2 x \sin (x) \cos (y)$

The below equation gives beautiful graph in desmos. I was trying to find a way to draw this graph using Javascript but for that I first need to express y in terms of x but I am not able to figure out ...
1 vote
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### Changing to polar coordinates bring the differential equation to $\Phi(\phi,\rho,\rho'(\phi)) = 0$ form.

I have a differential equation $y'=\dfrac{x+y}{x-y}$ and the problem says to change to polar coordinate system by assuming $x = \rho\cos\phi$, $y = \rho\sin\phi$ and $\rho=\rho(\phi)$ and bring the ...