Q.

Let y=y(x) be the solution of the differential equation, xy'-y=x2(xcosx+sinx),x>0 . If y(π)=π , then y''π2+yπ2 is equal to

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a

1+π2+π24

b

1+π2

c

2+π2

d

2+π2+π24

answer is C.

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Detailed Solution

y'-1xy=x(xcos+sinx)I.F. =e∫-1xdx=e-logx=x-1y·1x=∫1xx(xcosx+sinx)dx+Cyx=xsinx+Cput x=π. y(π)=πππ=πsinπ+C⇒C=1yx=xsinx+1y=x2sinx+xy(π2)=π24sinπ2+π2=π24+π2y'=x2cosx+sinx 2x+1 y"=x2(-sinx)+cosx 2x+2(sinx+xcosx) y"(π2)=-π24sinπ2+2π2cosπ2+2(sinπ2+π2cosπ2)           =-π24 +0+2+0 y(π2)+y"(π2)=π2+2
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Let y=y(x) be the solution of the differential equation, xy'-y=x2(xcosx+sinx),x>0 . If y(π)=π , then y''π2+yπ2 is equal to