Differential Equation Exact Solution

  • #1
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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
 
Last edited:

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
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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.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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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.
 

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