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General Solution of a Differential Equation

Verify that ...

Question

Verify that $x y = log y + c$ is a solution of the differential equation $(x y - 1) \frac{d y}{d x} + y^{2} = 0$ .

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Solution

Given $x y = l o g y + c$

Differentiate on both sides

$\Rightarrow x y^{^{'}} + y = \frac{1}{y} y^{^{'}}$

$\Rightarrow y^{^{'}} = \frac{y^{2}}{1 - x y}$

We will substitute $y^{^{'}}$ in given DE and we will verify

$\Rightarrow (x y - 1) \frac{y^{2}}{1 - x y} + y^{2}$

$\Rightarrow - y^{2} + y^{2} = 0 = R H S$

Given solution is correct for above differential equation

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