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By eliminating the arbitrary constant c, form the differential equation whose general solution is y = c*(1 - x²) + sqrt(1 - x²).
- (1 - x²)*dy/dx + 2*x*y = x*sqrt(1 - x²)
- (1 - x²)*dy/dx - 2*x*y = x*sqrt(1 - x²)
- (1 - x²)*dy/dx + 2*x*y = sqrt(1 - x²)
- (1 - x²)*dy/dx + x*y = x*sqrt(1 - x²)
Correct answer: (1 - x²)*dy/dx + 2*x*y = x*sqrt(1 - x²)
Solution
Differentiating y = c*(1 - x²) + sqrt(1 - x²) gives dy/dx in terms of c and x. From the original equation c = (y - sqrt(1 - x²))/(1 - x²). Substituting this c back and clearing denominators removes the constant and yields a first-order linear differential equation whose right side carries the sqrt(1 - x²) coming from the particular part of the solution.
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