Differential equationsVerify that the functions y_(1) and y_(2) are solutions
3 months ago
Verify that the functions y_(1) and y_(2) are solutions of the given differential equation. Do they constitute a fundamental set of solutions? (1-xcot x)y''-xy'+y=0, 0<x<pi ; y_(1)(x)=x, y_(2)(x)=sin x
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Basketballgirl Staff answered 3 months ago
We find the first and the second derivatives of y_(1) and y_(2): y'_(1)=1, y''_(1)=0, y'_(2)=cos x, y''_(2)=-sin x. We plug in those values into the differential equation: 0=(1-xcot x)y''_(1)-xy'_(1)+y_(1)=-x+x=0, 0=(1-xcot x)y''_(2)-xy'_(2)+y_(2)=(xcot x-1)sin x-xcos x+sin x =xcos x-sin x-xcos x+sin x=0. Thus, y_(1) and y_(2) are solutions of the given equation By Theorem 3.2.4, y_(1) and y_(2) form a fundamental set of solutions if and only if there is a point x where their Wro ian is nonzero. The wro ian is W(y_(1),y_(2))(x)=y_(1)(x)y'_(2)(x)-y_(2)(x)y'_(1)(x) =xcos x-sin x, which is nonzero for x=(pi)/(2) (for example). Therefore, y_(1) and y_(2) form a fundamental set of solutions of the given equation Result: Yes.
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