Let $y = y(x)$ be the solution of the differential equation $x^{4}dy + \left( 4x^{3}y + 2sinx \right

Question

Let $y = y(x)$ be the solution of the differential equation $x^{4}dy + \left( 4x^{3}y + 2sinx \right)dx = 0,x > 0,y\left( \frac{\pi}{2} \right) = 0$.

Then $\pi^{4}y\left( \frac{\pi}{3} \right)$ is equal to :

Accepted Answer

$\left( x^{4}dy + 4x^{3}ydx \right) = - 2sinxdx$

$${\Rightarrow \int d\left( x^{4}y \right) = \int - 2sinxdx }{\Rightarrow x^{4}y = 2cosx + c }{\Rightarrow x^{4}f(x) = 2cosx + c }$$As $f\left( \frac{\pi}{2} \right) = 0$

So, $c = 0$

$${\left( \frac{\pi}{3} \right)^{4}f\left( \frac{\pi}{3} \right) = 2cos\frac{\pi}{3} }{\pi^{4}f\left( \frac{\pi}{3} \right) = 81}$$

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