Let y=y(x) be the solution of the differential equation x d y d x - y = x 2 cot x , x ( 0 ,) . If y ( 2 ) = 2 , then 6 y (6 ) - 8 y ( 4 ) is equal to: [2026]

Question

Let y=y(x) be the solution of the differential equation $x \frac{d y}{d x} - y = x^{2} \cot x, x \in (0, π) .$ If $y (\frac{π}{2}) = \frac{π}{2},$ then $6 y (\frac{π}{6}) - 8 y (\frac{π}{4})$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

$x d y - y d x = x^{2} \cot x d x$

$x^{2} d (\frac{y}{x}) = x^{2} \cot x d x$

$d (\frac{y}{x}) = \cot x d x$

$\int d (\frac{y}{x}) = \int \cot x d x$

$\frac{y}{x} = \log_{e} (\sin x) + C$

$given y (\frac{π}{2}) = \frac{π}{2} \Rightarrow C = 1$

$y = x (\log_{e} (\sin x) + 1)$

$y (\frac{π}{6}) = \frac{π}{6} [- \log_{e} 2 + 1]$

$y (\frac{π}{4}) = \frac{π}{4} [- \frac{1}{2} \log_{e} 2 + 1]$

$6 y (\frac{π}{6}) - 8 y (\frac{π}{4})$

$= π [(- \log_{e} 2 + 1) + 2 (\frac{1}{2} \log_{e} 2 - 1)]$

$= π [1 - 2] = - π$

Solution

Q.
Let y=y(x) be the solution of the differential equation $x \frac{d y}{d x} - y = x^{2} \cot x, x \in (0, π) .$ If $y (\frac{π}{2}) = \frac{π}{2},$ then $6 y (\frac{π}{6}) - 8 y (\frac{π}{4})$ is equal to: [2026]

1 $π$

2 $3 π$

3 $- π$

4 $- 3 π$

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Solution

Q. Let y=y(x) be the solution of the differential equation xdydx-y=x2cotx, x∈(0,π). If y(π2)=π2, then 6y(π6)-8y(π4) is equal to: [2026]

1 π

2 3π

3 -π

4 -3π