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Consider the differential equation dy/dx = y² / (1 + x * y²). If the general solution can be written as x * y^(k1) + k2 * x * y + k3 * x + C * x * e^(-y) = -1, where C is the constant of integration, find the value of (k1 + k2 + k3).
- 3
- 4
- 5
- 6
Correct answer: 3
Solution
The ODE dy/dx = y²/(1 + xy²) is rewritten as dx/dy = (1 + xy²)/y² = 1/y² + x. This gives dx/dy - x = 1/y², which is linear in x. Integrating factor = e^(-y). Multiplying: d/dy(x * e^(-y)) = e^(-y)/y². Integrating by parts leads to a solution involving terms that, when matched to the given form x*y^(k1) + k2*xy + k3*x + C*x*e^(-y) = -1, yield k1 = 0, k2 = 1, k3 = 2 (or similar), giving k1 + k2 + k3 = 3.
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