CUET · MATHS · PYQ PAPER 2025
The particular solution of the differential equation \(\left\{x \sin ^2\left(\frac{y}{x}\right)-y\right\} d x+x d y=0, y=\frac{\pi}{4}\) when \(x=1\) is
- A \(x=e^{\tan ^2\left(\frac{y}{x}\right)+1}\)
- B \(x=e^{\tan ^2\left(\frac{y}{x}\right)-1}\)
- C \(x=e^{\cot \left(\frac{y}{x}\right)-1}\)
- D \(x=e^{\cot \left(\frac{y}{x}\right)+1}\)
Answer & Solution
Correct Answer
(C) \(x=e^{\cot \left(\frac{y}{x}\right)-1}\)
Step-by-step Solution
Detailed explanation
\( \frac{dy}{dx} = \frac{y}{x} - \sin^2\left(\frac{y}{x}\right) \) \( \text{Let } y=vx \implies v + x\frac{dv}{dx} = v - \sin^2(v) \) \( x\frac{dv}{dx} = -\sin^2(v) \implies \int \csc^2(v) dv = -\int \frac{dx}{x} \) \( -\cot(v) = -\ln|x| + C_1 \)…
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