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