Question

If $y^2 = ax^2 + bx + c$, where $a, b, c$ are constants, then $y^3 \frac{d^2y}{dx^2}$ is equal to

• A. A function of $y$
• B. function of both $x$ and $y$
• C. constant
• D. function of $x$

Hint:

Whenever you differentiate a quadratic implicit expression like $y^2 = P(x)$, the quantity $y^3 y''$ always eliminates $y$ entirely and simplifies to $\frac{4AC - B^2}{8}$ or a constant configuration matching the independent variable's constants.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Given y² = ax² + bx + c, determine the nature of the expression y³(d²y/dx²) through implicit differentiation.

Step 2: Key Formula or Approach:
Apply implicit differentiation twice using the chain rule and product rule, then substitute and simplify the resulting expression.

Step 3: Detailed Explanation:
First derivative: 2y(dy/dx) = 2ax + b → dy/dx = (2ax + b)/(2y). Differentiate 2y(dy/dx) = 2ax + b using the product rule: 2[y(d²y/dx²) + (dy/dx)²] = 2a → y(d²y/dx²) + (dy/dx)² = a. Multiply by y²: y³(d²y/dx²) + y²(dy/dx)² = ay². Note that y(dy/dx) = (2ax + b)/2, so y²(dy/dx)² = (2ax + b)²/4. Substituting and using y² = ax² + bx + c: y³(d²y/dx²) = a(ax² + bx + c) - (2ax + b)²/4. Expanding: a²x² + abx + ac - (4a²x² + 4abx + b²)/4 = ac - b²/4. This result depends solely on constants and is thus a constant function of x.

Step 4: Final Answer:
The expression yields a constant function of x, option (D).