Partial fraction with same denominator

Question

Is the following fraction (actually a Laplace transform) a kind of partial fraction?

$$\frac{4s+3}{{s^2}+3}$$

Can this be solved this way? $$\frac{A}{s}+\frac{B}{s+{\frac{3}{s}}}$$

If not can you please tell me how to find inverse transform?

Answer 1

If you want to keep everything real, this is already decomposed into partial fractions. For the inverse Laplace transform, just split it as $$ \frac{4s+3}{s^2+3} = 4 \frac{s}{s^2+3} + 3\frac{1}{s^2+3}. $$ You should be able to invert each term separately.

Answer 2

If you don't need to keep it real, the roots of $s^2 + 3$ are $\pm \sqrt{3} i$, and the partial fraction decomposition is $$\frac{4s+3}{s^2+3} = {\frac {2-i\sqrt {3}/2}{s-i\sqrt {3}}}+{\frac {2+i\sqrt {3}/2}{ s+i\sqrt {3}}}$$