JEE Mains · Maths · STD 12 - 9. differential equations
Let the solution curve \(y=y(x)\) of the differential equation \(\left(1+ e ^{2 x }\right)\left(\frac{ dy }{ dx }+ y \right)=1\) pass through the point \(\left(0, \frac{\pi}{2}\right)\). Then, \(\lim _{x \rightarrow \infty} e ^{x} y(x)\) is equal to.
- A \(\frac{\pi}{4}\)
- B \(\frac{3 \pi}{4}\)
- C \(\frac{\pi}{2}\)
- D \(\frac{3 \pi}{2}\)
Answer & Solution
Correct Answer
(B) \(\frac{3 \pi}{4}\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}+y=\frac{1}{1+e^{2 x}}\) So integrating factor is \(e^{\int 1 \cdot d x}=e^{x}\) So solution is \(y \cdot e^{x}=\tan ^{-1}\left(e^{x}\right)+c\) Now as curve is passing through \(\left(0, \frac{\pi}{2}\right)\) so \(c=\frac{\pi}{4}\)…
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