Is a solution of the form $f(x) = g(y)$ implicit?

Question

Given a solution of a variable separable equation, say the solution can't be expressed in the form $y = f(x)$ (given $y$ is the dependent variable) and is expressed as $f(x) = g(y)$. Then is it an implicit solution or an explicit solution?

Answer 1

I think you might have made a mistake in the notation; it should be $f(y)=g(x)$ where $f\not=g$. Still an implicit solution because you cannot compute $y$ in terms of $x$ only.

If you write $f(y)=f(x)$, then we may be able to express $y$ in terms of $x$. For example, if $f$ has an inverse, then taking the inverse function of both sides yields $y=x$.

However, if $f$ has no inverse, the solution is generally implicit. For example, the following has an explicit solution:

$$\begin{align} f(x) &= ax^2+bx+c\\ f(y)=f(x)\implies y&=\frac{-b\pm \sqrt{b^2 +4a(ax^2+bx)}}{2a}. \end{align}$$

Similarly, we can write explicit solutions for cases where $f(x)$ is a polynomial of order 3 or 4, but not order 5 or more. So in general, the solution is implicit.