How can I improve my understanding of solving differential equations?

[Beez]

Hello, I have tried to solve the following problem but did not succeed to do so.

[y(y^3 - x)]dx + [x (y^3 + x)]dy = 0
I sense that the key factor here is (y^3 - x ) and (y^3 +x), but could not figure out how to lead the equation to

dy/dx + P(x)y = Q(x) form.

The general answer for the problem is 2xy^3 - x^2 = Cy^2.

Once I can change the equation to dy/dx + P(x)y = Q(x) form, I can do the rest (probably anybody can...)

Thanks for your help in advance.

 

[himanshu121]

This is of the form f(xy)ydx +F(xy)xdy=0
here the Integrating factor is ::
$$\frac{1}{xy[f(xy)-F(xy)]}$$
and general Integral equation is ::
$$\int \frac{f(xy)+F(xy)}{f(xy)-F(xy)} \frac{d(xy)}{xy} + log\frac{x}{y} = c$$

 

[EvLer]

[y(y³ - x)]dx + [x (y³ + x)]dy = 0

that looks awfully a lot like exact DE: you'll have to get partial derivatives and the whole nine yards. Try that.

 

[himanshu121]

Well , there is an alternative if you use differentials
now
d(xy)= ydx +xdy
and
$$d(\frac{y}{x}) = \frac{xdy-ydx}{x^2}$$

Here u can rearrange ur diff eqn to

:: y³ d(xy) + x³d(y/x)=0
divide by y³ u will get the required answer after integrating

 

[Beez]

Exact D.E.

I believe it is an exact DE since I have just learned that part. But when I did
\partial M (x, y) / \partial y = 4y^3 - x and
\partial N (x, y) / \partial x = y^3 + 2x so they are not the same.

But I could not find integrating facutor to make their answers equal. What should I do now?

 

[Beez]

I will try that

Thank you "himanshu121". I will try that for now to see if I can understand that formula.


Well, I couldn't get it.

I have just started this differential equation class (independent). I thought I understood them well. However, when it comes to solve problems, I am experiencing a hard time. For example, this equation, I could not see why it Have some suggestion to improve my understanding?