The solution of equation ( 2 y - 1 ) ? d x - ( 2 x + 3 ) ? d y = 0 is:

Question

The solution of equation $(2 y - 1) d x - (2 x + 3) d y = 0$ is:

Smart Achievers · Accepted Answer

(3)

Ans. $\frac{2 x + 3}{2 y - 1} = k$

Explanation:

Given that,

$(2 y - 1) d x - (2 x + 3) d y = 0$

$\Rightarrow$ $(2 y - 1) d x = (2 x + 3) d y$

$\Rightarrow$ $\frac{d x}{2 x + 3} = \frac{d y}{2 y - 1}$

On integrating both sides, we get

$\frac{1}{2} \log (2 x + 3) = \frac{1}{2} \log (2 y - 1) + \log C$

$\Rightarrow$ $\frac{1}{2} [\log (2 x + 3) - \log (2 y - 1)] = \log C$

$\Rightarrow$ $\frac{1}{2} \log (\frac{2 x + 3}{2 y - 1}) = \log C$

$\Rightarrow$ ${(\frac{2 x + 3}{2 y - 1})}^{1 / 2} = C$

$\Rightarrow$ $\frac{2 x + 3}{2 y - 1} = C^{2}$

$\Rightarrow$ $\frac{2 x + 3}{2 y - 1} = k,$ $where k = C^{2}$

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