Let y = y ( x ) be the solution of the differential equation x 4 ? d y + ( 4 x 3 y + 2 sin x ) d x = 0 , x 0 , y ( 2 ) = 0 . Then 4 y ( 3 ) is equal to: [2026]

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Solution

Q.
Let $y = y (x)$ be the solution of the differential equation $x^{4} d y + (4 x^{3} y + 2 \sin x) d x = 0, x > 0,$ $y (\frac{π}{2}) = 0 .$ Then $π^{4} y (\frac{π}{3})$ is equal to: [2026]

1 81

2 72

3 92

4 64

Ans.
(1)

$(x^{4} d y + 4 x^{3} y d x) = - 2 \sin x d x$

$\Rightarrow \int d (x^{4} y) = \int - 2 \sin x d x$

$\Rightarrow x^{4} y = 2 \cos x + c$

$\Rightarrow x^{4} f (x) = 2 \cos x + c$

$As f (\frac{π}{2}) = 0$

$So, c = 0$

${(\frac{π}{3})}^{4} f (\frac{π}{3}) = 2 \cos \frac{π}{3}$

$π^{4} f (\frac{π}{3}) = 81$

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