Problem #$8$: $($a$)$ Given the initial value problem

$y^{\text{'}}+3y=10sint,y(0)=6,$

find the Laplace transform of the solution $Y(s).$

$($b$)$ Using the Laplace transform, $Y(s),$ find $y(t),$ the solution to our initial value problem.

$*$Don$\text{'}$t forget: Your answer should be a function of $t.$

$($A$)\frac{1}{s+3}-\frac{s-3}{s^2+1}$

$($B$)\frac{7}{s+3}-\frac{s-3}{s^2+1}$

$($C$)6+\frac{30}{s+3}+\frac{30}{s^2+1}$

$($D$)6+\frac{1}{s+3}-\frac{s-3}{s^2+1}$

$($E$)\frac{6}{s+3}-\frac{s}{s^2+1}$

$($F$)\frac{6}{s+3}+\frac{4}{s^2+1}$

$($G$)\frac{36}{s+3}+\frac{30}{s^2+1}$

$($H$)\frac{30}{s+3}+\frac{30}{s^2+1}$

Problem #$8($a$)$: $$ uarr Part $($a$)$ choices.

Enter your answer as a symbolic

Problem #$8($b$)$: $$ function of $t,$ as in these examples