In the text, I'm given with the following system of differential equations:
$$y_1'=-y_1+4y_2$$ $$y_2'=3y_1-2y_2$$
where $y_1(0)=3$ and $y_2(0)=4$
In the answer book, by doing Laplace transform it obtains the following system of differential equations:
$$(s-2)Y_1+3Y_2=1$$ $$-Y_1+(s+2)Y_2=0$$
How did it calculate this?
It's a typo. The correct LT for the system is as follows: \begin{align*} sY_1-y_1(0)&=-Y_1+4Y_2\\ sY_2-y_2(0)&=3Y_1-2Y_2\\ \\ sY_1-3&=-Y_1+4Y_2\\ sY_2-4&=3Y_1-2Y_2\\ \\ (s+1)Y_1-4Y_2&=3\\ -3Y_1+(s+2)Y_2&=4. \end{align*} Then continue.