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inverse of laplace 2/(x^2-36)-1/(x^2+6x)

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inverse laplace 2x²−36 −1x²+6x

Solution

e^6t−H(t)6

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L⁻¹{2x²−36 −1x²+6x }

Take the partial fraction of 2x²−36 : −16(x+6) +16(x−6)

=L⁻¹{−16(x+6) +16(x−6) −1x²+6x }

Take the partial fraction of 1x²+6x : 16x −16(x+6)

=L⁻¹{−16(x+6) +16(x−6) −16x +16(x+6) }

Use the linearity property of Inverse Laplace Transform:
For functions f(s), g(s) and constants a, b: L⁻¹{a·f(s)+b·g(s)}=a·L⁻¹{f(s)}+b·L⁻¹{g(s)}

=−L⁻¹{16(x+6) }+L⁻¹{16(x−6) }−L⁻¹{16x }+L⁻¹{16(x+6) }

L⁻¹{16(x+6) }: 16 e^−6t

L⁻¹{16(x−6) }: 16 e^6t

L⁻¹{16x }: 16 H(t)

=−16 e^−6t+16 e^6t−16 H(t)+16 e^−6t

Simplify −16 e^−6t+16 e^6t−16 H(t)+16 e^−6t: e^6t−H(t)6

=e^6t−H(t)6

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

inverse of laplace 2/(x^2-36)-1/(x^2+6x)

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