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

The general solution of the differential equation $(2 y-1) d x-(2 x+3) d y=0$ is
  1. $(2 x+3)^2=c(2 y-1)$
  2. $\frac{2 x+3}{2 y-1}=c$
  3. $(2 x+3)(2 y-1)=c$
  4. $(2 x+3)(2 y-1)^2=c$

Solution

$\begin{aligned} & (2 y-1) d x-(2 x+3) d y=0 \Rightarrow \frac{d y}{d x}=\frac{(2 y-1)}{(2 x+3)} \\ & \therefore \int \frac{d y}{2 y-1}=\int \frac{d x}{2 x+3} \\ & \frac{\log |2 y-1|}{2}=\frac{\log |2 x+3|}{2}+\log c_1 \Rightarrow \log \left|\frac{2 x+3}{2 y-1}\right|=-\log c_1=\log c \\ & \therefore \frac{2 x+3}{2 y-1}=c\end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 1)

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