JEE Mains · Maths · STD 12 - 9. differential equations
If \(y = y ( x )\) is the solution of the differential equation \(\frac{ dy }{ dx }+(\tan x ) y =\sin x , 0 \leq x \leq \frac{\pi}{3},\) with \(y (0)=0,\) then \(y \left(\frac{\pi}{4}\right)\) equal to :
- A \(\frac{1}{4} \log _{ e } 2\)
- B \(\left(\frac{1}{2 \sqrt{2}}\right) \log _{ e } 2\)
- C \(\log _{ e } 2\)
- D \(\frac{1}{2} \log _{ e } 2\)
Answer & Solution
Correct Answer
(B) \(\left(\frac{1}{2 \sqrt{2}}\right) \log _{ e } 2\)
Step-by-step Solution
Detailed explanation
\(\frac{d y}{d x}+(\tan x) y=\sin x ; 0 \leq x \leq \frac{\pi}{3}\) I.F. \(=e^{\int \tan x d x}=e^{\ell \sec x}=\sec x\) \(y \sec x=\int \tan x d x\) \(y \sec x=\int \tan x d x\) \(y \sec x=\ell n|\sec x|+C\) \(x=0, y=0 \quad \Rightarrow \quad \therefore c=0\)…
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