The degree of the differential equation whose solution is $y^2 = 8a(x+a)$, is
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When a parameter $a$ appears linearly as well as quadratically ($a^2$), substituting its first-derivative value back into the equation will inherently create a term containing $\left(\frac{dy}{dx}\right)^2$. Thus, you can instantly predict that the degree will be 2 without finishing the algebraic simplification!
Step 1: Understanding the Question: We are given the general solution y² = 8a(x+a) of a family of curves and must find the degree of the corresponding differential equation after eliminating the arbitrary constant a.
Step 2: Key Formula or Approach: Eliminate 'a' by differentiating the equation and substituting back. The degree is the highest power to which the highest-order derivative is raised after clearing radicals/fractions.
Step 3: Detailed Explanation: Differentiating gives 2y(dy/dx) = 8a → a = (y/4)(dy/dx). Substituting back and simplifying yields the differential equation containing (dy/dx)² as the highest power term, so the degree is 2.
Step 4: Final Answer: The degree of the differential equation is 2, matching option (A).
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