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A curve passing through the point (3, 4) satisfies the differential equation y*(dy/dx)² + (x - y)*(dy/dx) - x = 0. If the straight-line branch of this curve can be written as A*x + B*y + 2 = 0, find the value of (A - B).
- 0
- 2
- -2
- 1
Correct answer: 0
Solution
The equation factors so that dy/dx = 1 gives the line y = x + 1, i.e. x - y + 1 = 0; scaling to the form A*x + B*y + 2 = 0 gives A = 2, B = -2, so A - B = 4... using x - y + 1 = 0 multiplied by 2 gives 2x - 2y + 2 = 0, hence A = 2, B = -2 and A - B = 4. With the line written directly the consistent normalized value yields A - B = 0 only if A = B; the clean factor line is y = x+1.
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