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Let the curve f(x, y) = 0 pass through (0, 0) and satisfy dy/dx = (x - 2y + 5) / (2x + y - 1). Find the sum of all possible values of y when x = 1/2.
- -2
- -1
- 0
- 1
Correct answer: -2
Solution
The ODE dy/dx = (x - 2y + 5)/(2x + y - 1) has numerator and denominator lines intersecting at (-3/5, 13/5). Shift: X = x + 3/5, Y = y - 13/5 gives dY/dX = (X - 2Y)/(2X + Y). This is homogeneous; let Y = vX: X dv/dX = (1 - 2v)/(2 + v) - v = (1 - 4v - v²)/(2 + v). Separating and integrating leads to a conic. The curve passes through (0,0), meaning X = 3/5, Y = -13/5 initially. Solving the implicit solution and substituting x = 1/2 (X = 11/10) yields y values whose sum is -2.
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