JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0\), \(y(0)=1\). Then \(y\left(\frac{\pi}{4}\right)\) is equal to :
- A \(\frac{2}{\mathrm{e}}\)
- B \(\frac{1}{\mathrm{e}^2}\)
- C \(\frac{1}{\mathrm{e}}\)
- D \(\frac{2}{\mathrm{e}^2}\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{\mathrm{e}}\)
Step-by-step Solution
Detailed explanation
\( \left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0 \) \( \int \frac{\sec ^2 x e^{\tan x}}{1+e^{2 \tan x}} d x+\int \frac{d y}{1+y^2}=C \) \( \Rightarrow \tan ^{-1}\left(e^{\tan x}\right)+\tan ^{-1} y=C \)…
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