Question:
The general solution of the differential equation \((y^2 - x^3)\,dx - xy\,dy = 0\) \((x \ne 0)\) is : (where c is a constant of integration)
The general solution of the differential equation \((y^2 - x^3)\,dx - xy\,dy = 0\) \((x \ne 0)\) is : (where c is a constant of integration)
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Put \(v = y^2\); the equation becomes a linear differential equation in v.
Updated On: Nov 11, 2026
- \(y^2 + 2x^3 + cx^2 = 0\)
- \(y^2 + 2x^2 + cx^3 = 0\)
- \(y^2 - 2x + cx^3 = 0\)
- \(y^2 - 2x^3 + cx^2 = 0\)
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The Correct Option is A
Solution and Explanation
\(\frac{dy}{dx} = \frac{y^2 - x^3}{xy}\). Multiply by \(2y\): \(\frac{d(y^2)}{dx} = \frac{2y^2}{x} - 2x^2\).
Let \(v = y^2\): \(\frac{dv}{dx} - \frac{2}{x}v = -2x^2\).
Integrating factor \(= e^{-2\ln x} = x^{-2}\).
\(\frac{d}{dx}\left(\frac{v}{x^2}\right) = -2 \Rightarrow \frac{v}{x^2} = -2x + k\).
\(y^2 = -2x^3 + kx^2 \Rightarrow y^2 + 2x^3 - kx^2 = 0\); with \(c = -k\): \(y^2 + 2x^3 + cx^2 = 0\).
Answer: (1) \(y^2 + 2x^3 + cx^2 = 0\)
Let \(v = y^2\): \(\frac{dv}{dx} - \frac{2}{x}v = -2x^2\).
Integrating factor \(= e^{-2\ln x} = x^{-2}\).
\(\frac{d}{dx}\left(\frac{v}{x^2}\right) = -2 \Rightarrow \frac{v}{x^2} = -2x + k\).
\(y^2 = -2x^3 + kx^2 \Rightarrow y^2 + 2x^3 - kx^2 = 0\); with \(c = -k\): \(y^2 + 2x^3 + cx^2 = 0\).
Answer: (1) \(y^2 + 2x^3 + cx^2 = 0\)
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