Let y=y(x) be the solution of the differential equation x^4dy+(4x^3y+2 x)dx=0,x0,y(( )/(2…

Question

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

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

Accepted Answer

$\begin{gathered} {x}^{4}dy+(4{x}^{3}y+2\sin x)dx=0 \ \frac{dy}{dx}+\frac{4{x}^{3}}{{x}^{4}}y=\frac{-2\sin x}{{x}^{4}} \ \frac{dy}{dx}+\frac{4}{x}y=\frac{-2\sin x}{{x}^{4}} \ ={e}^{\int \frac{4}{x}dx={e}^{\ln {x}^{4}}={x}^{4}} \ {x}^{4}\frac{dy}{dx}+4{x}^{3}y=-2\sin x \ \frac{d}{dx}({x}^{4}y)=-2\sin x \ \Rightarrow {x}^{4}y=2\cos x+c \ y(\frac{\pi }{2})=0 \ \Rightarrow c=0 \ 	herefore {x}^{4}y=\cos x \ \Rightarrow y=\frac{2\cos x}{{x}^{4}} \end{gathered}$

$\begin{gathered} y(\frac{\pi }{3})=\frac{2\cos (\frac{\pi }{3})}{{(\frac{\pi }{3})}^{4}}=\frac{1}{{(\frac{\pi }{3})}^{4}}=\frac{81}{{\pi }^{4}} \ {\pi }^{4}y(\frac{\pi }{3})=81 \end{gathered}$

Solutions · 1