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

User Calebe
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1 Answer

4 votes

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.
User Peter Bushnell
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