Project ForgeEngineering Mathematics
Differential Equations
GATE 2011
Hard
Which ONE of the following choices is a solution of the differential equation dy/dx=y²/x+y/x−2/x? (c is a real constant)
- Ay=(c−x²)/(c+2x²)
- By=(c+2x²)/(c−x²)
- Cy=(c−x³)/(c+2x³)
- y=(c+2x³)/(c−x³)
Explanation
The RHS factors as (y+2)(y−1)/x. Direct substitution of y=(c+2x³)/(c−x³) gives dy/dx=9cx²/(c−x³)², while computing (y+2)(y−1)/x from the same expression gives [3c/(c−x³)]×[3x³/(c−x³)]/x=9cx²/(c−x³)² — an exact match, confirming this is a valid solution.
Source: GATE CH 2011 (2011)
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