Question

The equation y=xsinx and wants to prove x dpower 3y/dxpower3+x dy/dx+2y=0

Answer 1

`y=x\sin x`


`(\mathrm dy)/(\mathrm dx)=x(\mathrm d(\sin x))/(\mathrm dx)+(\mathrm dx)/(\mathrm dx)\sin x=x\cos x+\sin x`


`(\mathrm d^2y)/(\mathrm dx^2)=x(\mathrm d(\cos x))/(\mathrm dx)+(\mathrm dx)/(\mathrm dx)\cos x+(\mathrm d(\sin x))/(\mathrm dx)`

`(\mathrm d^2y)/(\mathrm dx^2)=-x\sin x+\cos x+\cos x`

`(\mathrm d^2y)/(\mathrm dx^2)=-x\sin x+2\cos x`


`(\mathrm d^3y)/(\mathrm dx^3)=-\left(x(\mathrm d(\sin x))/(\mathrm dx)+(\mathrm dx)/(\mathrm dx)\sin x\right)+2(\mathrm d(\cos x))/(\mathrm dx)`

`(\mathrm d^3y)/(\mathrm dx^3)=-x\cos x-\sin x-2\sin x`

`(\mathrm d^3y)/(\mathrm dx^3)=-x\cos x-3\sin x`

Substitute the derivatives into the given ODE and check to see if the relation is an identity.


`x(\mathrm d^3y)/(\mathrm dx^3)+x(\mathrm dy)/(\mathrm dx)+2y=0`

`x(-x\cos x-3\sin x)+x(x\cos x+\sin x)+2x\sin x=0`

`-x^2\cos x-3x\sin x+x^2\cos x+x\sin x+2x\sin x=0`

`0=0`

and we're done.