CUET · MATHS · PYQ PAPER 2025
If \(e^y=\log x\), then which of the following is true?
- A \(x \frac{d^2 y}{d x^2}-\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)
- B \(\frac{d^2 y}{d x^2}-x \frac{d y}{d x}=0\)
- C \(\frac{d^2 y}{d x^2}-\left(\frac{d y}{d x}\right)^2+1=0\)
- D \(x \frac{d^2 y}{d x^2}+x\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)
Answer & Solution
Correct Answer
(D) \(x \frac{d^2 y}{d x^2}+x\left(\frac{d y}{d x}\right)^2+\frac{d y}{d x}=0\)
Step-by-step Solution
Detailed explanation
\(e^y \frac{dy}{dx} = \frac{1}{x}\) \(\frac{dy}{dx} = \frac{1}{x e^y}\) \(e^y \left(\frac{dy}{dx}\right)^2 + e^y \frac{d^2 y}{dx^2} = -\frac{1}{x^2}\) \(\frac{d^2 y}{dx^2} = -\frac{1}{x^2 e^y} - \frac{1}{x^2 e^{2y}} = -\frac{e^y+1}{x^2 e^{2y}}\)…
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