AP EAMCET · Maths · Differential Equations
The general solution of the differential equation \((2 x-y)^2 d y-2(2 x-y)^2 d x-2 d x=0\) is
- A \(\log (2 x-y)=2 x+c\)
- B \((2 x-y)^3+4 y=c\)
- C \((2 x-y)^3+6 x=c\)
- D \(\log (2 x-y)=2 y+c\)
Answer & Solution
Correct Answer
(C) \((2 x-y)^3+6 x=c\)
Step-by-step Solution
Detailed explanation
\((2 x-y)^2 d y = [2(2 x-y)^2 + 2] d x\) \(\frac{d y}{d x} = 2 + \frac{2}{(2 x-y)^2}\) Let \(v = 2 x-y \implies \frac{d v}{d x} = 2 - \frac{d y}{d x}\) \(2 - \frac{d v}{d x} = 2 + \frac{2}{v^2}\) \(v^2 d v = -2 d x\) \(\int v^2 d v = \int -2 d x\) \(\frac{v^3}{3} = -2x + c_1\)…
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