Solve using Laplace Transformations
1. $\displaystyle y' + y = sin(x)$ $\displaystyle where$ $\displaystyle y(0) = 1$
$\displaystyle sY(s) - y(0) + 4Y(s) = 1/(s^2 + 1)$
$\displaystyle Y(s)(s + 4) = 1/(s^2 + 1) - 1$
$\displaystyle Y(s) = (s^2 + 2)/[(s + 4)(s^2 +1)]$
2. $\displaystyle y^(^3^) + y' = e^x $ $\displaystyle where$ $\displaystyle y(0)=y'(0)=y"(0)=0$
$\displaystyle s^3Y(s) - s^2y(0) - sy'(0) - y"(0) + Y(s) - y(0) = 1/(s - 1)$
$\displaystyle Y(s)(s^3 + s) = 1/(s - 1)$
$\displaystyle Y(s) = 1/[x(x + 1)(x^2 + 1)]$
I'd greatly appreciate it if someone could assist me in finishing these problems as I'm unsure of what to do.