Project ForgeEngineering Mathematics
Calculus
GATE 2022
Medium
For the exact differential equation, du/dx = −xu²/(2+x²u), which one of the following is the solution?
- Au² + 2x² = constant
- Bxu² + u = constant
- (1/2)x²u² + 2u = constant
- D(1/2)ux² + 2x = constant
Explanation
Rewrite as x²u du + xu² dx + 2du = 0. Since d[(1/2)x²u²] = xu²dx + x²u du (product rule applied to x²u²), the first two terms combine into d[(1/2)x²u²], and the last term is d(2u). So the equation is exactly d[(1/2)x²u² + 2u] = 0, giving (1/2)x²u²+2u=constant. Verified by implicit differentiation: differentiating option (C) reproduces the given ODE exactly.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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