What are x and y in a differential equation like dy/dx = f(x)g(y)?

Key takeaway: This question comes up a lot when students first meet separable differential equations. The notation looks compact, but each symbol has a job.

In a differential equation such as

$\frac{dy}{dx}=f(x)g(y),$

think of it like this:

• $x$ is the independent variable.
• $y$ is the dependent variable, usually written as a function of $x$, such as $y=y(x)$.
• $\frac{dy}{dx}$ means the derivative of that function with respect to $x$.

So yes, $y$ depends on $x$.

What does $g(y)$ mean?

It means the value of the function depends on the current value of $y$, not directly on $x$. Since $y$ itself changes with $x$, the term $g(y)$ changes as $x$ changes too.

That is why the equation is called separable:

$\frac{dy}{dx}=f(x)g(y)$

can be rearranged into

$\frac{1}{g(y)}\,dy=f(x)\,dx.$

At that point, you integrate both sides:

$\int \frac{1}{g(y)}\,dy=\int f(x)\,dx.$

A simple way to picture it

• $f(x)$ uses the input $x$
• $g(y)$ uses the output value $y$, which is itself changing with $x$

So the equation links how fast $y$ changes to both the current position $x$ and the current value $y$.

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Pitfalls the pros know 👇 A frequent mix-up is thinking that $g(y)$ means "the same y as in $dy/dx$" in a separate sense. It is the same dependent variable, but now it is being fed into another function. Also, don’t treat $dy$ and $dx$ as ordinary numbers too early; the rearrangement works here because the equation is separable.

What if the problem changes? If the equation were

$\frac{dy}{dx}=f(x),$

then $y$ would still be a function of $x$, but the rate of change would depend only on $x$. If instead it were

$\frac{dy}{dx}=g(y),$

then the change rate depends only on $y$. In both cases, $y$ is still the unknown function you are solving for.

Tags: dependent variable, separable differential equation, initial value problem