Differential Equations

We know that $\mathcal{L}\{t\} = \frac{1}{s^2}$. By the first shifting property (frequency shifting theorem): $\mathcal{L}\{e^{at} f(t)\} = F(s - a)$ Here with $a = 3$ and $f(t) = t$: $\mathcal{L}\{t e^{3t}\} = \frac{1}{(s - 3)^2}$ Comparing with $\frac{1}{(s - a)^b}$: $a = 3$ and $b = 2$. Hence, $a + b = 3 + 2 = 5$.