JEE Mains · Maths · STD 12 - 9. differential equations
Let \(g\) be a differentiable function such that \(\int_0^x g(t) d t=x-\int_0^x \operatorname{tg}(t) d t, x \geq 0\) and let \(y=y(x)\) satisfy the differential equation \(\frac{d y}{d x}-y \tan x=\) \(2(x+1) \sec x g(x), x \in\left[0, \frac{\pi}{2}\right)\). If \(y(0)=0\), then \(y\left(\frac{\pi}{3}\right)\) is equal to
- A \(\frac{2 \pi}{3 \sqrt{3}}\)
- B \(\frac{4 \pi}{3}\)
- C \(\frac{2 \pi}{3}\)
- D \(\frac{4 \pi}{3 \sqrt{3}}\)
Answer & Solution
Correct Answer
(B) \(\frac{4 \pi}{3}\)
Step-by-step Solution
Detailed explanation
Diff. w.r.t. x \(\begin{aligned} & g(x)=1-x g(x) \\ & g(x)=\frac{1}{1+x} \\ & \text { so } \frac{d y}{d x}-y \tan x=2 \sec x \\ & I F=e^{-\int \tan d x}=e^{\log \cos x}=\cos x \end{aligned}\) solution of D.E.…
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