JEE Mains · Maths · STD 12 - 9. differential equations
If \(y = y ( x )\) is the solution of the differential equation \(\left(1+ e ^{2 x }\right) \frac{ dy }{ dx }+2\left(1+ y ^{2}\right) e ^{ x }=0\) and \(y (0)=0\), then \(6\left( y ^{\prime}(0)+\left( y \left(\log _{ e } \sqrt{3}\right)\right)^{2}\right)\) is equal to
- A \(2\)
- B \(-2\)
- C \(-4\)
- D \(-1\)
Answer & Solution
Correct Answer
(C) \(-4\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{1+y^{2}}+\frac{2 e^{x}}{1+e^{2 x}} d x=0\) \(\dots(i)\) on integration \(\tan ^{-1} y+2 \tan ^{-1} e^{x}=c\) \(\because y(0)=0\) so, \(C =\frac{\pi}{2} \Rightarrow \tan ^{-1} y +2 \tan ^{-1} e ^{ x }=\frac{\pi}{4}\) from eq.\((i)\),…
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