Question

What is the general solution of the differential equation? `y^2 \ dx + xy \ dy = 0`

What can you say about the solutions?

Answer 1

Solution of this differential euation was `1/2x^2y^2=C`

Explanation:

`M(x, y)=y^2` and `N(x,y)=xy`

Once, I controlled condition exactness of differential equation,

`(dM)/dy=2y` and `(dN)/dx=y`

Hence, it wasn't exact.

I decided to integration factor accordingly `x`

Hence,

`P=e^[int ((dM)/dy-(dN)/dx)/N*dx`

=`e^[int (2y-y)/(xy)*dx`

=`e^[int (dx)/x]`

=`e^(Lnx)`

=`x`

After multiplying both sides with `P=x`,

`xy^2*dx+x^2y*dy=0`

After controlling exactness condition of it,

`(d(xy^2))/dy=(d(x^2y))/dx=2xy`

Thus, solution of it,

`xy^2*dx+x^2y*dy=0`

`d(1/2x^2y^2)=0`

`1/2x^2y^2=C`

Answer 2

Multiplying by `x` we get an exact equation:

`xy^2 \ dx + x^2y \ dy = 0`

Whose solution is:

`y = +- C/x`

ie a family of hyperbolas.

Explanation:

`y^2 \ dx + xy \ dy = 0 ..... [A]`

Suppose we have:

`M(x,y) \ dx + N(x,y) \ dy = 0`

Then the DE is exact if `M_y-N_x=0`

`M = y^2 => M_y = 2y`
`N= xy => N_x = y`

`M_y - N_x != 0 =>` Not an exact DE

So, we seek an Integrating Factor `mu(u)` such that

`(muM)_y = (muN)_x`

So, we compute::

`I=(M_y-N_x)/N = (2y-y)/(xy) = 1/x`

So the Integrating Factor is given by:

`mu(x) = e^(int \ I \ dx)`
`\ \ \ \ \ \ \ = e^(int \ 1/x \ dx)`
`\ \ \ \ \ \ \ = e^(lnx)`
`\ \ \ \ \ \ \ = x`

So when we multiply the DE [A] by the IF we now get an exact equation:

`(x)y^2 \ dx + (x)xy \ dy = 0 .`

`xy^2 \ dx + x^2y \ dy = 0`

And so if we redefine the function `M` and `N`:

`M = xy^2 => M_y = 2xy`
`N= x^2y => N_x = 2xy`

Making the equation exact

Then, our solution is given by:

`f_x = M` and `f_y = N` and

If we consider `f_y = N`, then:

`f = int \ x^2y \ partial y + g(x)`, where we treat `x` as constant
`\ \ = (x^2y^2)/2 + g(x)`

And now we consider `f_x = M` and we differentiate the last result to get:

`f_x = xy^2 + g'(x)`

As `f_x=M` then we have:

`xy^2 + g'(x) = xy^2`
`:. g'(x) = 0 => g(x) = K`

Leading to the GS:

`(x^2y^2)/2 = K => (xy)^2=c`

`:. xy=+- C => y = +- C/x`