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**TheEmptySet** $\displaystyle s^4X-s^3(0)-s^2(0)-s(2)-0+s^3X-s^2(0)-s(0)-2=\frac{1}{s^2+1}$

$\displaystyle s^3(s+1)X=2s+2+\frac{1}{s^2+1} \iff X=\frac{2}{s^3}+\frac{1}{s^3(s+1)(s^2+1)}=\frac{2s ^2+2s^2+2s+3}{s^3(s+1)(s^2+1)}$

So the partial fractions will look like

$\displaystyle \frac{A}{s}+\frac{B}{s^2}+\frac{C}{s^3}+\frac{D}{s +1}+\frac{Es+F}{s^2+1}=\frac{2s^2+2s^2+2s+3}{s^3(s +1)(s^2+1)}$

Multiplying out and solving gives

$\displaystyle A=0,B=-1,C=3,D=-\frac{1}{2},E=\frac{1}{2},F=\frac{1}{2}$