solve rthe differential equation (2x^2-y^2)dx-xydy=0

Question

its either a homogeneous equation or an exact differential  equation am not sure which one is it

Answer 1

dy/dx

Answer 2

=1/xy•(2x^2 - y^2)

Answer 3

= 2-(y/x)^2/(y/x)

Answer 4

( y / x )

Answer 5

→ F(x) :

Answer 6

y'= xF'+F ,

Answer 7

XF'=2-F^2/F - F

Answer 8

→ FdF/1-F^2

Answer 9

= 2 dx/x ,

Answer 10

-1/2• d(1-F^2)/1-F^2

Answer 11

=2dx/x

Answer 12

→(1-F^2)-1 • d (1-F^2)

Answer 13

=-4/x dx,

Answer 14

Ln(1-F^2)

Answer 15

=-4 Lnx + c power ~

Answer 16

→ (1-F^2)= c/x^4....,

Answer 17

1-(y/x)^2= c/x^4

Answer 18

→ y=√x^2-c/x^2

Answer 19

(2x^2-y^2)dx=xydy; dy/dx=(2x^2-y^2)/xy=2x/y-y/x.

Let u=y/x=yx^-1, dy/dx=2/u-u, du/dx=(1/x)dy/dx-y/x^2, so xdu/dx=dy/dx-y/x, and dy/dx=xdu/dx+y/x=xdu/dx+u=2/u-u.

xdu/dx=2/u-2u=2(1/u-u)=2(1-u^2)/u.

integral(udu/(1-u^2))=integral(dx/x).

Therefore, -(1/2)ln|1-u^2|=ln|ax|, where a is a constant. This can be written: ln|ax|+(1/2)ln|1-u^2|=(1/2)ln|a^2x^2(1-u^2)|=0, thus ln|a^2x^2(1-u^2)|=0.

So, raising e to each side: a^2x^2(1-u^2)=1, 1-u^2=1/a^2x^2, u^2=1-1/a^2x^2; replacing u by y/x:

y^2=x^2-1/a^2 and y=sqrt(x^2-A^2). (Constant A replaces 1/a.)