Why Does the dy Disappear When Solving Differential Equations?

[Physics-Pure]

Hello all~

Given the equation:
dy/dx = (x/y)
I know we would initially go to:
∫dy =∫ (x/y) dx
then too:
∫(y)(dy) = ∫x dx
Until arriving at:
(y²/2) + C₁ = (x²/2) + C₂
(y²) - (x²) = C

My question is:
Where does the dy disappear to in step 4? Where the anti-derivative is taken.

Why does ∫dy become just y when solving an equation of the form
dy/dx = (x² + 1), but it disappears in the first example?


Thank you~

 

[arildno]

This is, basically, wrong all the way.
First off, if you have dy/dx=x/y(x), we may rewrite this, bu multiplying both sides with y(x) as:

y(x)dy/dx=x.

Then, up to an arbitrary constant of integration, we'll have.

int((y(x)dy/dx)dx)=int(xdx).
That is integrating BOTH sides with respect to the same variable, i.e, "x".

Now, on the left-hand side, we use the reverse of the chain rule of differentiation, that is, integration by substitution, letting "y" be our integration variable.
There is no magical disappearance of any variables or infinitesemals.

 

[Physics-Pure]

This is, basically, wrong all the way.
First off, if you have dy/dx=x/y(x), we may rewrite this, bu multiplying both sides with y(x) as:

y(x)dy/dx=x.Now, on the left-hand side, we use the reverse of the chain rule of differentiation, that is, integration by substitution, letting "y" be our integration variable.
There is no magical disappearance of any variables or infinitesemals.

Where did that y(x) come from?
And are you saying that we let y = y(x)dy/dx)dx?

P.S. I was simply following the "Introduction to differential equations" video, under calculus. Found here: http://www.hippocampus.org/Calculus & Advanced Math;jsessionid=BAEE0BB1E88F4A594768EEBE4D8FC1EA

 

[Studiot]

Why does ∫dy become just y when solving an equation of the form

Following the method presented (which is known as variables separable or separation of variables)

The answer is that ∫dy is not ∫dy it is the ∫1dy.
When you integrate this the integral of 1 is y and the dy drops out as it did in the previous example. You are not integrating the dy.
You asked why the dy drops out - well it is really a book- keeping symbol I see someone else already told you this in another thread.

You should note that the answer to your first question

Where does the dy disappear to in step 4?

is that is disappears to the same place the dx disappears to in the same line and for the same book-keeping reason.

 

[blue_raver22]

Hello there!

I understand your confusion about the disappearance of dy in step 4. When taking the anti-derivative, we are essentially "undoing" the derivative operation. In other words, we are finding the original function that, when differentiated, would give us the given equation. In this case, the original function is y, so the dy disappears after integrating.

In the second example, ∫dy becomes just y because we are integrating with respect to y, not x. Therefore, we treat y as a constant and it does not disappear during integration. In the first example, we are integrating with respect to x, so we treat y as a variable and it disappears after integrating.

I hope this helps clarify your question. Keep up the good work with differential equations!