Verify that the functions y1 and y2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions? (1−xcot⁡x)y″−xy′+y=0,0&lt;x&lt;π;y1(x)=x,y2(x)=sin⁡x

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Verify that the functions y1 and y2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions? (1−xcot⁡x)y″−xy′+y=0,0&amp;lt;x&amp;lt;π;y1(x)=x,y2(x)=sin⁡x

Branson Grimes · Accepted Answer

We find the first and the second derivatives of y1 and y2:y1′=1,y″1=0,y2′=cos⁡x,y″2=−sin⁡x.We plug in those values into the differential equation:0=(1−xcot⁡x)y″1−xy1′+y1=−x+x=0,0=(1−xcot⁡x)y″2−xy2′+y2=(xcot⁡x−1)sin⁡x−xcos⁡x+sin⁡x=xcos⁡x−sin⁡x−xcos⁡x+sin⁡x=0.Thus,y1 and y2 are solutions of the given equationBy Theorem 3.2.4, y1 and y2 form a fundamental set of solutions if and only if there is a point x where their Wroian is nonzero. The wroian isW(y1,y2)(x)=y1(x)y2′(x)−y2(x)y1′(x)=xcos⁡x−sin⁡x,which is nonzero for x=π2 (for example). Therefore,y1 and y2 form a fundamental set of solutions of the given equationResult:Yes.

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-x cot x)y''-xy'+y=0, 0<x<pi ; y_1 (x)=x, y_2 (x)=sin x

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