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Question
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Options
y − x3 = 2cx
2y − x3 = cx
2y + x2 = 2cx
y + x2 = 2cx
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Solution
2y − x3 = cx
We have,
\[x\frac{dy}{dx} - y = x^2\]
\[\Rightarrow \frac{dy}{dx} - \frac{1}{x}y = x^2 \]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = - \frac{1}{x} \]
\[Q = x^2 \]
Now,
\[I . F . = e^{- \int\frac{1}{x}dx} = e^{- \log\left| x \right|} \]
\[ = e^{log\left| \frac{1}{x} \right|} \]
\[ = \frac{1}{x}\]
\[y \times I . F = \int x^2 \times I . Fdx + C\]
\[ \Rightarrow y\frac{1}{x} = \int x^2 \times \frac{1}{x}dx + C\]
\[ \Rightarrow y\frac{1}{x} = \int xdx + C\]
\[ \Rightarrow y\frac{1}{x} = \frac{x^2}{2} + C\]
\[ \Rightarrow 2y - x^3 = Cx\]
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