Question

Can someone please help!!! How do you solve for the inverse laplace of 1/(s(s+4)(s+4))

Answer 1

@Spacelimbus

Answer 2

\[\Large \frac{1}{s(s+4)^2} \]?

Answer 3

yes

Answer 4

Partial fraction decomposition

\[\Large \frac{1}{s(s+4)^2} = \frac{A}{s}+ \frac{B}{s+4}+ \frac{C}{(s+4)^2}\]

Answer 5

so you'd get A(S+4)(S+4)^2 +B(s)(S+4)^2 +C(s)(S+4)^2

Answer 6

If you multiply through you are left with

\[\Large 1=A(s+4)^2+B(s+4)+Cs \]

Answer 7

I thought you are supposed to multiply A by the values under B and C?

Answer 8

I made a mistake above, but no you want to solve for the coefficients so you might get an easier expression

Answer 9

\[ \Large 1=A(s+4)^2+Bs(s+4)+Cs\]

Answer 10

better now it seems

Answer 11

wouldnt you get A=0, B=0 and C=-1/4

Answer 12

Sorry for the mistake again, I messed up the above form, so if you solved for that your answer will be wrong.

Answer 13

\[\Large 1=(A+B)s^2 +(8A+4B+C)s+16A \]

Answer 14

could you please explain how you go that

Answer 15

\[A+B=0 \\ 8A+4B+C=0
\\
16A=0 \]

Answer 16

Yes, I multiplied the above form by s(s+4)^2 (make sure you cancel on the right hand side right) and then you factor likewise terms. When you did that, you can set both sides of the equation equal, they are equal if and only if their coefficients match.

Answer 17

\[\Large \frac{1}{s(s+4)^2} = \frac{A}{s}+ \frac{B}{s+4}+ \frac{C}{(s+4)^2} \]
General setup of partial fraction decomposition, multiply through by \(s(s+4)^2\), it will disappear entirely on the left hand side of the equation and piecewise on the righthandside of the equation.

Answer 18

Then expand the righthand side and factor likewise terms, set the coefficients equal to the coefficients on the lefthand side to make a valid equation out of it LHS=RHS, in this case a lot of the coefficients have to be zero, because there is only 1 on the LHS.

Answer 19

you will end up with a 3x3 system of linear equations.

\[A+B=0 \\ 8A+4B+C=0
\\
16A=1 \]

*corrected the last line*

Answer 20

okay so far?

Answer 21

sorry im still confused on how you got the equation

Answer 22

can you please show me by the work?

Answer 23

The idea of partial fraction decomposition is that you find a different way of writing the quotient, so in the end there will be an easier form for it which is especially useful in integral calculus and for LaPlace Transformations.

The idea is that you completely factor the denominator and then you split it up, so each new quotient becomes its own denominator