JEE Mains · Maths · STD 12 - 9. differential equations
Let \(x = x(y)\) be the solution of the differential equation \(2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0\), \(y > 1\), \(x(e) = e\). Then \(x(e^2)\) is equal to:
- A \(\dfrac{3}{2} e^2\)
- B \(\dfrac{2}{3} e^2\)
- C \(e^2\)
- D \(2e^2\)
Answer & Solution
Correct Answer
(B) \(\dfrac{2}{3} e^2\)
Step-by-step Solution
Detailed explanation
The given differential equation is \(2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0\). Dividing by \(2y^2\), we get: \(\dfrac{dx}{dy} - \dfrac{x}{y} + \dfrac{x^2}{2y^2} = 0\) This is a homogeneous differential equation. Let \(x = vy\), then \(\dfrac{dx}{dy} = v + y \dfrac{dv}{dy}\).…
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