JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(\begin{array}{l} \cos x(3 \sin x+\cos x+3) d y= (1+y \sin x(3 \sin x+\cos x+3)) d x \end{array}\) ; \(0 \leq x \leq \frac{\pi}{2}, y(0)=0 .\) Then \(, y\left(\frac{\pi}{3}\right)\) is equal to ..... .
- A \(2 \log _{e}\left(\frac{2 \sqrt{3}+9}{6}\right)\)
- B \(2 \log _{e}\left(\frac{2 \sqrt{3}+10}{11}\right)\)
- C \(2 \log _{e}\left(\frac{\sqrt{3}+7}{2}\right)\)
- D \(2 \log _{ e }\left(\frac{3 \sqrt{3}-8}{4}\right)\)
Answer & Solution
Correct Answer
(B) \(2 \log _{e}\left(\frac{2 \sqrt{3}+10}{11}\right)\)
Step-by-step Solution
Detailed explanation
\(\cos x(3 \sin x+\cos x+3) d y\) \(\begin{array}{l}=(1+y \sin x(3 \sin x+\cos x+3)) d x \\ \frac{d y}{d x}-(\tan x) y=\frac{1}{(3 \sin x+\cos x+3) \cos x}\end{array}\) \(I.F. =e^{\int-\tan x d x}=e^{\ell n|\cos x|}=|\cos x|\)…
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