Differential of a y mixed with x

[Karol]

Homework Statement

Find dy of $~xy^2+x^2y=4$

Homework Equations

Differential of a product:
$$d(uv)=u\cdot dv+v\cdot du$$

The Attempt at a Solution

$$2xy~dy+y^2~dx+x^2~dy+2xy~dx=0$$
$$x(2y+x)dy=-(y+2x)dx$$

 

[ehild]

Homework Statement

Find dy of $~xy^2+x^2y=4$

Homework Equations

Differential of a product:
$$d(uv)=u\cdot dv+v\cdot du$$

The Attempt at a Solution

$$2xy~dy+y^2~dx+x^2~dy+2xy~dx=0$$
$$x(2y+x)dy=-(y+2x)dx$$

You miss a y on the RHS.

 

[Karol]

$$x(2y+x)dy=-y(y+2x)dx$$
$$\frac{dy}{y}=-\frac{y+2x}{2y+x}~\frac{dx}{x}$$
Is there a meaning for $~\frac{dy}{y}~$?

 

[WWGD]

$$x(2y+x)dy=-y(y+2x)dx$$
$$\frac{dy}{y}=-\frac{y+2x}{2y+x}~\frac{dx}{x}$$
Is there a meaning for $~\frac{dy}{y}~$?

What type of meaning are you considering? You can see it as $\frac{1}{y} dy$.

 

[Karol]

And what is $~\frac{1}{y} dy~$? what do i do with it? $~d(ln~y)=\frac{1}{y} dy$
But the book teaches logs only later

 

[scottdave]

I think to solve your original problem, you want to rearrange it so that you have (dy/dx) = {something}

 

[Karol]

I have to find $~\frac{dy}{dx}=f(x)~$, but i have $~\frac{dy}{dx}=f(x,y)$

 

[WWGD]

I would guess this is a case of implicit differentiation, where you assume y is a function of x. Is that in your book?

 

[Karol]

Yes, y=f(x) only

 

[WWGD]

Then differentiate the whole expression as a function of x, using the chain rule on y=y(x).

 

[Karol]

Then differentiate the whole expression as a function of x, using the chain rule on y=y(x).

$$2xy\frac{dy}{dx}+y^2+x^2\frac{dy}{dx}+2xy=0$$
$$dy=-\frac{y^2+2xy}{x^2+2xy}~dx$$

 

[WWGD]

$$2xy\frac{dy}{dx}+y^2+x^2\frac{dy}{dx}+2xy=0$$
$$dy=-\frac{y^2+2xy}{x^2+2xy}~dx$$

Good, but no need to split the dy/dx. Leave it as a single unit and solve for it.

 

[Karol]

Good, but no need to split the dy/dx. Leave it as a single unit and solve for it.

$$\frac{dy}{dx}=-\frac{y^2+2xy}{x^2+2xy}$$
What i can do, and i can't even that, is to integrate, but it's not the point and the book didn't teach it yet.
What else can i do with a derivative which involves y?
I need dy, that is what i was asked, and any dy=f(x)dx
I need to express, find, what is y=y(x)

 

[WWGD]

$$\frac{dy}{dx}=-\frac{y^2+2xy}{x^2+2xy}$$
What i can do, and i can't even that, is to integrate, but it's not the point and the book didn't teach it yet.
What else can i do with a derivative which involves y?
I need dy, that is what i was asked, and any dy=f(x)dx
I need to express, find, what is y=y(x)

Sorry, then do split into dy and dx parts.I am not clear, just what is your goal here, to find y?

 

[Karol]

just what is your goal here, to find y?

I was asked to find dy

 

[WWGD]

I was asked to find dy

My apologies then, do separate dy and dx and solve for dy.

 

[Karol]

Now i see that the book's answer is like mine, leaving y undevelopped

 

[WWGD]

Now i see that the book's answer is like mine, leaving y undevelopped

" undeveloped"? How so? What is the book's answer?

 

[Karol]

" undeveloped"? How so? What is the book's answer?

 

[WWGD]

View attachment 205470

I see, thanks.

 

[Karol]

Thank you scottdave and WWGD