Solve the initial$-$value problem. $5$y$^(\text{'}\text{'})+7$y$^(\text{'})+2$y$=0$ y$(0)=7$ y$^(\text{'})(0)=-4$ y$($x$)=$

$($Note: I feel like I am doing something wrong with the isolation of C$1$ & C$2.$ Please be explicit about each arithmetic step to reach the solution. e$.$g$.,$ Don't write $"........(1)"$ because I am trying to understand what I am doing wrong in the vague land of $"......")$