MHB Why can y(x) be rewritten as just y?

[find_the_fun]

In an example problem we start with [math]y'(x)=3y(x)[/math]. The next step in solving for y is [math]\frac{dy}{dx}=3y[/math] how can you drop the (x) part? I'm not used to seeing y written with something after the parentheses, I thought y is used because it's easier than writing f(x) which means a function named f with the argument x.

 

[Chris L T521]

Just as $f(x)$ is a function with argument $x$, $y(x)$ also denotes a function with argument $x$. So instead of saying $\displaystyle\frac{dy}{dx}(x) = 3y(x)$ or $y^{\prime}(x)=3y(x)$, it's cleaner to write it as $\displaystyle\frac{dy}{dx} = 3y$ or $y^{\prime}=3y$ because it should be clear from context that we're working with functions of $x$.

I hope this clarifies things!

 

[find_the_fun]

Just as $f(x)$ is a function with argument $x$, $y(x)$ also denotes a function with argument $x$. So instead of saying $\displaystyle\frac{dy}{dx}(x) = 3y(x)$ or $y^{\prime}(x)=3y(x)$, it's cleaner to write it as $\displaystyle\frac{dy}{dx} = 3y$ or $y^{\prime}=3y$ because it should be clear from context that we're working with functions of $x$.

I hope this clarifies things!

Is this a necessary step or is it just to make the writing look cleaner?

 

[Ackbach]

Is this a necessary step or is it just to make the writing look cleaner?

The second. I would add that anything that doesn't sacrifice clarity and makes the writing easier is a good thing, since all mathematicians are lazy and strive for economy of effort (hence the Einstein summation convention, which some point to as the greatest invention since sliced bread, simply because it saved a lot of writing!).