Differential Equation Exact Solution

[jesuslovesu]

Homework Statement

(2x - y) + (2y-x) dy/dx = 0
y(1) = 3
Solve and determine where the solution is approximately valid.

Homework Equations

The Attempt at a Solution

Ux = 2x - y
Uy = 2y - x

U = x^2 - yx + h(y)
Uy = 2y -x = -x + h'(y)
h'(y) = 2y
h(y) = y^2

x^2 - yx + y^2 = c
1 - 3 + 3^2 = c = 7

I'm not really sure where to go from here,
x^2 - yx + y^2 = 7
However, the answer is quite a bit different:
y = x + sqrt( 28 - 3x^2)/2 and I can't quite see the step toward that. (x-y)^2 = 7-xy but I still can't seem to solve for y

 

[quasar987]

I have only read the last line of your post, so I don't know if your steps are right, but regarding solving for y in (x-y)^2 = 7-xy, notice that solving this equation for y(x) is the same as finding the roots of the quadratic equation y^2 + (-x)y + (x^2-7) = 0.

 

[HallsofIvy]

And, in this case "b²- 4ac"= (-x)²- 4(1)(x[sup[2]- 7)= x²- 4x²+ 28= 28- 3x². That's where that came from.