Hello
Can anyone please help me and tell me if I have done the inverse laplace transform on this equation correctly?

See attached word document

\frac{1}{5s}+\frac{s+2}{5(s^{2}+2s+5)}

What you do is complete the square in the denominator.

\frac{s+2}{5(s^{2}+2s+5)}

=\frac{s}{5(s^{2}+2s+5)}+\frac{2}{5(s^{2}+2s+5)}

s^{2}+2s+5=(s+1)^{2}+4


Also, \frac{1}{s}=1, so \frac{1}{5s}=\frac{1}{5}

It should be:

L^{-1}(s)=\frac{1}{5}e^{-t}cos(2t)+\frac{1}{10}e^{-t}sin(2t)+\frac{1}{5}

y' - y = te^2t