Solve the given differential equation by undetermined coefficients. y''' - 6y'' = 4 - cos(x).

Solution:

Given differential equation y''' - 6y'' = 4 - cos(x).

Assume dy/dx=m

Then characteristic equation of this differential becomes: m³-6m²=0 or m²⋅(m-6)=0

Hence roots of it m₁=m₂=0 and m₃=6

General Solution: y₁=Ae⁶ˣ+Bx+C

particular solution must be in form: yp=Dx² + Esinx + Fcosx

y' = 2Dx + Ecosx - Fsinx, y'' = 2D - Esinx - FcosX and y''' = -Ecosx + Fsinx

(-Ecosx + Fsinx) -6(2D - Esinx - Fcosx) = 4 - cosx

-12D + (F + 6E) sinx + (6F - E)cosx = 4 - cosx

Equate the coefficients, we get

-12D = 4 ; F + 6E = 0 ; 6F - E= -1

D= -1/3; E= 1/37; F= -6/37

So, particular Solution: yp= -1/3x² + 1/37sinx -6/37cosx

Hence, total solution for the given D.E is y= y₁ + yp

y= Ae⁶ˣ + Bx + C -(1/3)x² + (1/37)sinx - (6/37)cosx

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Solve the given differential equation by undetermined coefficients. y''' - 6y'' = 4 - cos(x).

Summary:

By solving the given differential equation by undetermined coefficients. y''' - 6y'' = 4 - cos(x), we get y= Ae⁶ˣ + Bx +C -(1/3)x² + (1/37)sinx − (6/37)cosx.