Inverse Laplace Transformations

[jimmypoopins]

Homework Statement

Find the inverse Laplace transform of the given functions:
3. $$\frac{2}{s^2+3s-4}$$

7. $$\frac{2s+1}{s^2-2s+2}$$

Homework Equations

Inverse Laplace Transform Table

The Attempt at a Solution

on 3. i made the denominator look like $$(s+4)(s-1)$$ but i got lost from there. i couldn't find anything on the table resembling the equation.

on 7. i completed the square and got $$\frac{2s+1}{(s-1)^2+1}$$, which resembles $$\frac{s-a}{(s-a)^2+b^2}$$ on the table, yet i can't seem to make the numerator look like $$(s-a)$$.

if it helps anyone, the answers are 3. $$\frac{2}{5}e^t-\frac{2}{5}e^{-4}$$ and 7. $$2e^{t}cos(t)+3e^{t}sin(t)$$.

thanks in advance.edit: p.s. if anyone has a better table of inverse laplace transformations, the one i posted is really hard to read. I'm using the one in my book but it might be useful, if you need to refer to one in a table when trying to help, to refer to a table you're more comfortable reading.

 

[Dick]

For the first one, use partial fractions to write the problem as A/(s-1)+B/(s+4) (find the constants A and B). Break up the second fraction as well. Write it as 2(s-1)/((s-1)^2+1) +C/((s-1)^2+1) (find the constant C). Use the linearity of the laplace transform.

 

[jimmypoopins]

For the first one, use partial fractions to write the problem as A/(s-1)+B/(s+4) (find the constants A and B). Break up the second fraction as well. Write it as 2(s-1)/((s-1)^2+1) +C/((s-1)^2+1) (find the constant C). Use the linearity of the laplace transform.

thank you very much. I'm really bad with partial fractions, so i guess my brain sort of tried to not think of that method. i guess i'll touch up on them because this chapter is filled with inverse laplace transformations.