NDA 2024 · PYQ · Differential Equations / Variable Separation · medium
If dy/dx = 2eˣ y³, y(0) = 1/2 then what is 4y²(2 - eˣ) equal to?
- A.1✓ Correct
- B.2
- C.3
- D.4
Separating variables: dy/y³ = 2eˣ dx. Integrating: -1/(2y²) = 2eˣ + C. At x = 0, y = 1/2, so -1/(2·1/4) = 2 + C, i.e., -2 = 2 + C, so C = -4. Thus -1/(2y²) = 2eˣ - 4, so 1/(2y²) = 4 - 2eˣ = 2(2 - eˣ). Therefore 1/y² = 4(2 - eˣ), giving 4y²(2 - eˣ) = 1.
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