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Find the equation of the curve that passes through the point (a,-1/a) and satisfies the differential equation
y - x(dy)/(dx) = a (y² + (dy)/(dx)).
- (x + a)(1 + ay) = -4a²y
- (x + a)(1 - ay) = 4a²y
- (c + a)(1 - ay) = -4a²y
- None of these
Correct answer: (c + a)(1 - ay) = -4a²y
Solution
From y - x y' = a(y^2 + y') we get y'(x+a)=y - a y^2, separable as dy/(y(1-ay)) = dx/(x+a). Integrating and using the point (a,-1/a) yields the relation (x+a)(1-ay) = -4 a^2 y (option 3 with c a typo for x).
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