MHT CET · Maths · Differential Equations
The general solution of the differential equation \((2 y-1) d x-(2 x+3) d y=0\) is
- A \((2 x+3)^2=c(2 y-1)\)
- B \(\frac{2 x+3}{2 y-1}=c\)
- C \((2 x+3)(2 y-1)=c\)
- D \((2 x+3)(2 y-1)^2=c\)
Answer & Solution
Correct Answer
(B) \(\frac{2 x+3}{2 y-1}=c\)
Step-by-step Solution
Detailed explanation
\((2 y-1) d x-(2 x+3) d y=0 \Rightarrow \frac{d y}{d x}=\frac{(2 y-1)}{(2 x+3)}\)
\(\therefore \int \frac{d y}{2 y-1}=\int \frac{d x}{2 x+3}\)
\(\frac{\log |2 y-1|}{2}=\frac{\log |2 x+3|}{2}+\log c_1 \Rightarrow \log \left|\frac{2 x+3}{2 y-1}\right|=\) \(-\log c_1=\log c\)
\(\therefore \frac{2 x+3}{2 y-1}=c\)
\(\therefore \int \frac{d y}{2 y-1}=\int \frac{d x}{2 x+3}\)
\(\frac{\log |2 y-1|}{2}=\frac{\log |2 x+3|}{2}+\log c_1 \Rightarrow \log \left|\frac{2 x+3}{2 y-1}\right|=\) \(-\log c_1=\log c\)
\(\therefore \frac{2 x+3}{2 y-1}=c\)
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