Differential Equation : y/x substitution
I used a v = y/x substitution on this and I got a really weird looking answer so I just want to see how right or wrong I am.
Solve: (x^2 + 2y^2)dx + xydx = 0
v = y/x and dy/dx = v +x(dv/dx)
So: x(dv/dx) = (-1/v) - v
after integrating I get:
(-1/2)ln|1 + v^2| = ln(x) + C
subbing back gives
(1 + y^2/x^2)^2 = x + e^c
Can someone please check this work and then come up with an equation for y? I have an answer but it looks so weird I don't think its right. Thanks.
(y/x) sub difEq corrections made...please help me finish?
I have seen the errors and redid it. If you don't mind I would like to stay with the y/x sub instead of the Bernoulli sub since we have not yet covered them ok?
Solve: (x^2 + 2y^2)dx + xydy = 0
xydy = -(x^2 + 2y^2)dx
dy/dx = -(x/y) - (2y/x)
let v = y/x, y = vx, dy/dx = v + xdv/dx
v + x(dv/dx) = -(1/v) - 2v
please check algebra here:
x(dv/dx) = (-1-3v^2)/v
-vdv/(1+3v^2) = dx/x
with a u substitution I get
LHS: -(1/6)ln|1 + 3y^2/x^2|
ln|1 + 3y^2/x^2| = -6ln|x| + C
Is this correct so far? Also please could you show me some skills on how to wrap this up correctly. Thanks so much.