Which is the Correct Answer for a Separable Differential Equation?

[gingermom]

Homework Statement

A separable differential equation is a first-order differential equation that can be algebraically manipulated to look like:
a. f(x)dx +f(y)dx
b. f(y)dy = g(x)dx
c. f(x)dx = f(y)dy
d. g(y)dx = f(x)dx
e. both f(y)dy=g(x)dx and f(x)dx = f(y)dy

Homework Equations

The Attempt at a Solution

B is the way it is defined in the book so I assume that is the answer, but "e" gave me pause. I feel like the two equations in "e" are not the same but I can not explain why they are different. I have the feeling knowing that would help fill in some of the pieces between memorizing how to do this stuff and really understanding it. Then again if I am off and the answer may be "e". Can anyone shed some light?

 

[Mark44]

Homework Statement

A separable differential equation is a first-order differential equation that can be algebraically manipulated to look like:
a. f(x)dx +f(y)dx
b. f(y)dy = g(x)dx
c. f(x)dx = f(y)dy
d. g(y)dx = f(x)dx
e. both f(y)dy=g(x)dx and f(x)dx = f(y)dy

Homework Equations

The Attempt at a Solution

B is the way it is defined in the book so I assume that is the answer, but "e" gave me pause. I feel like the two equations in "e" are not the same but I can not explain why they are different. I have the feeling knowing that would help fill in some of the pieces between memorizing how to do this stuff and really understanding it. Then again if I am off and the answer may be "e". Can anyone shed some light?

It looks to me like b, c, and d are reasonable answers. Separating a differential equation entails getting a function involving x (for example) and dx on one side, and another function (not necessarily different) of y (for example) and dy on the other side.

As an example of c), you might get an equation you start with separated to x²dx = y²dy. Integrate to get x³ = y³ + C. The choices for b and d are essentially the same, just with different variables. Choice a is not an equation, so would be out of the running.

This question looks like it might have been written by an instructor in a hurry...

 

[gingermom]

Maybe I am not as confused as I thought. I did double check to make sure I typed it exactly as it was on the worksheet. Thanks for responding.

 

[Mark44]

Also, one could choose e as well.

BTW, I posted in the other question you asked, and that should get you through that problem.

 

[LCKurtz]

I disagree. I think the only answer is (b.). If you are giving a definition of a separable DE, you wouldn't require the coefficients of dy and dx to be the same function such as in (c.) and (e.). It's true they are separable, but they aren't the definition.

(a.) and (d.) are out because one of them isn't an equation and the other doesn't have a dy.

 

[Mark44]

I disagree. I think the only answer is (b.). If you are giving a definition of a separable DE, you wouldn't require the coefficients of dy and dx to be the same function such as in (c.) and (e.). It's true they are separable, but they aren't the definition.

But I don't think the problem is defining a separable equation, only that a separable equation can be put into one of the forms a. through e.

(a.) and (d.) are out because one of them isn't an equation and the other doesn't have a dy.

I missed that d. had dx on both sides.

 

[gingermom]

I think you are correct, the answer is B and this was the difference he was looking for that I couldn't put my finger on. While both of the ones in E maybe separable, it is not the definition. Exactly what a question is trying to ascertain is always clearer to the person who writes it.

 

[LCKurtz]

I think you are correct, the answer is B and this was the difference he was looking for that I couldn't put my finger on. While both of the ones in E maybe separable, it is not the definition. Exactly what a question is trying to ascertain is always clearer to the person who writes it.

Yes. Admittedly the question is poorly worded. But if he wanted the other interpretation surely the question would have been phrased "Which of these equations are separable?".

 

[LCKurtz]

But I don't think the problem is defining a separable equation, only that a separable equation can be put into one of the forms a. through e.

Try putting c in that statement:

A separable equation can be put into form c.

Don't you agree that is false?

 

[Mark44]

I'm not saying that every separable DE can be put into form c. My thinking was that if we had this DE: $\frac{x^2}{y^2} = \frac{dy}{dx}$, it is certainly separable, and we could manipulate it to become x²dx = y²dy. This equation has the form f(x)dx = f(y)dy.

 

[LCKurtz]

I'm not saying that every separable DE can be put into form c. My thinking was that if we had this DE: $\frac{x^2}{y^2} = \frac{dy}{dx}$, it is certainly separable, and we could manipulate it to become x²dx = y²dy. This equation has the form f(x)dx = f(y)dy.

We can just agree to disagree. I don't think your interpretation of the question is valid.

 

[Mark44]

I agree that b is a much more likely answer than c. Also, choosing b doesn't rule out the possibility that f $\equiv$ g, which would cover choice c.

 

[gingermom]

Just for the record - the correct choice was indeed B. Thanks for the discussion. I had never really thought about what having g(x) versus f(x) meant in that definition. Too often texts and teachers gloss over those little details that really helps with understanding. That is why I love this forum. Just wish I had found it earlier.