solve this d²y/dx² = sin x
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The given DE can be written as [D²+ 1]y = sinx . It is a linear differential equation with constant coefficients . To solve it, we have it’s A.E. equation as : (m²+1) = 0 ==> m = ± i , therefore, its C.F. is y = Acosx + B sinx , where A, B are arbitrary constants . And P.I. = sinx/(D²+1) = x.sinx/2D = (1/2)∫x.sinx dx
= (1/2)[-x.cosx + ∫cosx dx]
= (1/2)[-x cosx + sinx] . Therefore, complete solution to the given DE is ;
y = C.F. + P.I.
= Acosx + Bsinx +(1/2).(sinx - xcosx) or
= A cosx + (B+1/2)sinx - (1/2)x.cosx
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