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The differential equation dy/dx = (ax + h) / (by + k) represents a parabola. Under which condition(s) on a and b does this hold?
- a = -2, b = 0
- a = -2, b = 2
- a = 0, b = 2
- a = 0, b = 0
Correct answer: a = 0, b = 2
Solution
Separating variables: (by + k)dy = (ax + h)dx. Integrating: b*y²/2 + k*y = a*x²/2 + h*x + C. This is a general second-degree curve. For a parabola, one of the squared terms must be missing but not both: either a = 0 (leaving only y² term — parabola opening along x-axis) or b = 0 (leaving only x² term — parabola opening along y-axis). Both a=0 and b=0 gives a line, not a parabola. Options: a=0, b=2 (parabola in y) and a=-2, b=0 (parabola in x) are both valid; a=-2, b=2 gives an ellipse/hyperbola; a=0, b=0 gives linear (degenerate).
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