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

1 Answers

Best Answer

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.