The solution of the homogeneous differential equation represents which family of geometric curves?

Answer and explanation

Correct answer: A. Circles passing through the origin with centers on the -axis

  1. Step 1: Substitute , so . The equation becomes: .
  2. Step 2: Simplify: .
  3. Step 3: Integrate: . Substitute : . This is a family of circles passing through the origin with centers at on the -axis.
Answer
Circles passing through the origin with centers on the -axis.
Why not the others: Circles with centers on the -axis would have equation ; Parabolas have equations ; Rectangular hyperbolas have equations .

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