Question

Cho hàm số \(y=x\sin x\)

a) Chứng minh rằng:

y^’’’ + y^’ +2sinx = 0

b) Tìm x khi y^’’ + y =2

c) Tính y⁽⁵⁾(\(\frac{\pi}{2}\))

Answer 1

\(y'=sinx+xcosx\)

\(y''=cosx+cosx-xsinx=2cosx-xsinx\)

\(y'''=-2sinx-sinx-xcosx=-3sinx-xcosx\)

\(y^{\left(4\right)}=-3cosx-cosx+xsinx=-4cosx+xsinx\)

\(y^{\left(5\right)}=4sinx+sinx+xcosx=5sinx+xcosx\)

a/

\(y'''+y'+2sinx=-3sinx-xcosx+sinx+xcosx+2sinx=0\)

b/ \(y''+y=2\Leftrightarrow2cosx-xsinx+xsinx=2\)

\(\Leftrightarrow cosx=1\Rightarrow x=k2\pi\)

c/ \(y^{\left(5\right)}\left(\frac{\pi}{2}\right)=5sin\frac{\pi}{2}+\frac{\pi}{2}.cos\frac{\pi}{2}=5\)