Discussion
The differential equation corresponding to the family of curves :
\[ y = c(x - c)^{2}, \]
where c is an arbitrary constant, is :
Answer: B
The differential equation is \((y')^{3} = 4y(xy' - 2y)\).
From \(y = c(x - c)^{2}\), \(y' = 2c(x - c)\). Check the right-hand side of option B:
\[ \begin{aligned} xy' - 2y &= 2cx(x - c) - 2c(x - c)^{2} \\ &= 2c(x - c)[x - (x - c)] \\ &= 2c^{2}(x - c) \end{aligned} \]
\[ \begin{aligned} 4y(xy' - 2y) &= 4c(x - c)^{2} \cdot 2c^{2}(x - c) \\ &= 8c^{3}(x - c)^{3} = (y')^{3} \end{aligned} \]
The constant c is eliminated, as it must be for the family's differential equation. The other options fail the same substitution. For option D, for instance, \((y')^{2} = 4c^{2}(x - c)^{2}\), while \(3y(xy' + 2y) = 6c^{2}(x - c)^{3}(2x - c)\), a polynomial of higher degree in x.
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