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

[find_the_fun]

In an example problem we start with \(\displaystyle y'(x)=3y(x)\). The next step in solving for y is \(\displaystyle \frac{dy}{dx}=3y\) 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!).

 

[blue_raver22]

In mathematics, we often use shorthand notation to represent functions. The notation y(x) is commonly used to represent a function y with the argument x. This notation is used for convenience and does not change the meaning of the function. Therefore, y(x) and y are interchangeable and can be used interchangeably in equations.

In the example problem, we start with y'(x)=3y(x). This can be rewritten as y'=3y, as the (x) part is not necessary for understanding the function. Similarly, in the next step, we have \frac{dy}{dx}=3y. Again, the (x) part can be dropped as it does not change the meaning of the function.

It is important to note that the notation y(x) does not mean that y is multiplied by x. It simply represents a function y with the argument x. This notation is commonly used because it is shorter and easier to write than f(x), which represents a function named f with the argument x.

In summary, y(x) can be rewritten as y because they represent the same function and the (x) part is not necessary for understanding the function. Shorthand notation is commonly used in mathematics and should not be a cause for confusion.