JEE Mains · Maths · STD 12 - 9. differential equations
If the solution curve \(y=y(x)\) of the differential equation \(\left(1+y^2\right)\left(1+\log _6 x\right) d x+x d y=0, x>0\) passes through the point \((1,1)\) and \(y(\mathrm{e})=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}\), then \(\alpha+2 \beta\) is ...........
- A \(4\)
- B \(3\)
- C \(8\)
- D \(10\)
Answer & Solution
Correct Answer
(B) \(3\)
Step-by-step Solution
Detailed explanation
\(\int\left(\frac{1}{x}+\frac{\ln x}{x}\right) d x+\int \frac{d y}{1+y^2}=0 \) \( \ln x+\frac{(\ln x)^2}{2}+\tan ^{-1} y=C\) Put \(x=y=1\) \( \therefore C=\frac{\pi}{4} \) \( \Rightarrow \ln x+\frac{(\ln x)^2}{2}+\tan ^{-1} y=\frac{\pi}{4}\) Put \(\mathrm{x}=\mathrm{e}\)…
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