Let y = y(x) be the solution of the differential equation

$\frac{d y}{d x} + \frac{\sqrt[]{2} y}{2 c o s^{4} x - c o s 2 x} = x e^{t a n^{- 1} (\sqrt[]{2} c o t 2 x)},$ $0 < x < \frac{π}{2} w i t h y (\frac{π}{4}) = \frac{π^{2}}{32} .$

9 Views | Posted 2 months ago

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alok kumar singh | Contributor-Level 10

2 months ago

2cos⁴ x – cos²x = $2 {(\frac{1 + c o s 2 x}{2})}^{2} - c o s 2 x$

$= \frac{1 + c o s^{2} 2 x}{2}$

$\sqrt[]{2} \int \frac{2 d x}{1 + c o s^{2} 2 x} = \sqrt[]{2} \int 2 \cdot \frac{s e c^{2} 2 x d x}{2 + t a n^{2} 2 x}$

$= \sqrt[]{2} \cdot \frac{1}{\sqrt[]{2}} \cdot t a n^{- 1} (\frac{t a n x}{\sqrt[]{2}})$

now IF = $e^{\int P d x}$

$e^{\sqrt[]{2} \int 2 \frac{s e c^{2} 2 x d x}{2 + t a n^{2} 2 x}} = e^{t a n^{- 1} (\frac{t a n 2 x}{\sqrt[]{2}})}$

Solution: $y \cdot e^{t a n^{- 1} (\frac{t a n 2 x}{\sqrt[]{2}})} = \int x . e^{t a n^{- 1} (\sqrt[]{2} \cdot c o t 2 x)} \cdot e^{t a n^{- 1} (\frac{t a n 2 x}{\sqrt[]{2}})} d x$

$y \cdot e^{t a n^{- 1} (\frac{t a n 2 x}{\sqrt[]{2}})} = \frac{x^{2}}{2} \cdot e^{π / 2} + C$

at $x = \frac{π}{4}, y = \frac{π^{2}}{32}, C = 0$

at $x = \frac{π}{3}, y = \frac{π^{2}}{18} e^{t a n^{- 1} α}$

$y \cdot e^{t a n^{- 1}} \frac{(t a n 2 \frac{π}{3})}{\sqrt[]{2}} = \frac{π^{2}}{18} \cdot e^{π / 2}$

$\frac{π^{2}}{18} \cdot e^{t a n^{- 1} α} \cdot e^{t a n^{- 1}} (- \frac{\sqrt[]{3}}{2}) = \frac{π^{2}}{18} \cdot e^{π / 2}$

tan^-1 a + tan^-1 $(- \sqrt[]{3} / 2) = \frac{π}{2}$

cot^-1a = tan^-1 $(- \frac{\sqrt[]{3}}{2})$

tan^-1 $\frac{1}{α} = t a n^{- 1} (- \frac{\sqrt[]{3}}{2})$

$\frac{1}{α} = - \sqrt[]{3} / 2$

2 = $\frac{2}{3}$

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Let y = y(x) be the solution of the differential equation

$\frac{d y}{d x} + \frac{\sqrt[]{2} y}{2 c o s^{4} x - c o s 2 x} = x e^{t a n^{- 1} (\sqrt[]{2} c o t 2 x)},$ $0 < x < \frac{π}{2} w i t h y (\frac{π}{4}) = \frac{π^{2}}{32} .$

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Let y = y(x) be the solution of the differential equation dydx+2y2cos4x−cos2x=xetan−1(2cot2x), 0<x<π2with y(π4)=π232.

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Let y = y(x) be the solution of the differential equation

$\frac{d y}{d x} + \frac{\sqrt[]{2} y}{2 c o s^{4} x - c o s 2 x} = x e^{t a n^{- 1} (\sqrt[]{2} c o t 2 x)},$ $0 < x < \frac{π}{2} w i t h y (\frac{π}{4}) = \frac{π^{2}}{32} .$