Let y₁(x) and y₂(x) be two solutions of the differential equation dy/dx = x. If y₁(0) = 0 and y₂(0) = 4, then what is the number of points of intersection of the curves y₁(x) and y₂(x)?
A.No point✓ Correct
B.One point
C.Two points
D.More than two points
Explanation
The general solution of dy/dx = x is y = x²/2 + C. With y₁(0) = 0, C₁ = 0, so y₁ = x²/2. With y₂(0) = 4, C₂ = 4, so y₂ = x²/2 + 4. Setting y₁ = y₂: x²/2 = x²/2 + 4 ⟹ 0 = 4, which has no solution. So the curves never intersect — no points of intersection.
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