What i want to ask is both question 20 and 21 with Laplace transforms
Actually i can solve question 20 and 21 with method of undetermined coefficients but . In laplace transform, i don't know how to solve them 'efficiently'(especially 21, it can't be partial fraction expansion. So i dont know how to solve)
For 21 take a Laplace transform to obtain $$(s^2F(s)-s-1)+(sF(s)-1)+F(s)=\frac2{s^3}$$ which you can rearrange to obtain $$F(s)=\frac{s^4+2s^3+2}{s^5+s^4+s^3}$$
can you take it from here?
PS. See this page.
Addendum
we have $$F(s)=\frac2{s^3}-\frac2{s^2}+\frac{s+\frac12}{s^2+s+1}+\frac72\frac1{s^2+s+1}$$