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Step-by-step explanation:
this is proved that left hand side = right hand side
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y' = 6+ e^3x.b + 3.e^3x.(a+bx)
y" = e^3x.3b + 9.e^3x.(a+bx) + 3.e^3x.b
= 3be^3x + 9e^3x.(a+bx) + 3be^3x
y" - 6y'
= 3be^3x + 9e^3x.(a+bx) +3be^3x -36 - 6be^3x- 18e^3x.(a+bx)
= 9e^3x.(a+bx) - 18e^3x.(a+bx) - 36
= - 9e^3x.(a+bx) - 36
y" - 6y +9y
= -9e^3x.(a+bx) - 36 + 54(x+1) +9(a+bx) e^3x
= -36 + 54(x+1)
= 54x + 18
(proved)
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