MATH 20D Lecture 24

is called irreducible, i.e. we cannot write it as for real numbers

Example. Find the inverse Laplace transform of $Y(s)=\frac{2s^2-s+3}{(s^2+1)(s-2)}$

Partial fraction decomp


When :
When :
When :
Partial Fraction Decomposition
So where
and
Thus,

Find $\mathcal{L}\{\frac{2s-3}{s^2+2s+10}\}$

Complete the square from denom - from this,
The denominator suggests that the inverse Laplace transform will be a combination of and whose Laplace transforms are and
We let


So
Thus,

Find the inverse Laplace transform of $Y(s)=\frac{12}{(s^2+2s+5{)}^2}$

Look at the denom , which suggests
So the answer will involve and
suggests that the answer also involves and
In fact,

and
Next we write as a linear combo of and solve for the coefficients