Question:
The solution of differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y\) is
The solution of differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y\) is
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Treat \(x\) as the dependent variable: the equation becomes linear in \(x\) with respect to \(y\).
Updated On: Nov 12, 2026
- \(x=y^{2}+c\)
- \(y=x^{2}+c\)
- \(x=y\left(y^{2}+c\right)\)
- \(y=x\left(x^{2}+c\right)\)
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The Correct Option is C
Solution and Explanation
Rewrite: \(\frac{dx}{dy}=\frac{x+2y^3}{y}\), i.e. \(\frac{dx}{dy}-\frac{1}{y}x=2y^2\).
Integrating factor \(=e^{\int -\frac{1}{y}dy}=\frac{1}{y}\).
\(\frac{x}{y}=\int 2y^2\cdot\frac{1}{y}\,dy=\int 2y\,dy=y^2+c\).
Hence \(x=y(y^2+c)\).
Answer: (3) \(x=y\left(y^{2}+c\right)\)
Integrating factor \(=e^{\int -\frac{1}{y}dy}=\frac{1}{y}\).
\(\frac{x}{y}=\int 2y^2\cdot\frac{1}{y}\,dy=\int 2y\,dy=y^2+c\).
Hence \(x=y(y^2+c)\).
Answer: (3) \(x=y\left(y^{2}+c\right)\)
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