To finish this let's just take the x(0) = 3 condition first...
This says that for this equation below.
x(t) = C1 . [1 , -i/2] + C2 . [1 , i/2]
When you set t=0, the 'x parts' of the LHS side should equal 3.
By 'x parts' I mean the following...
Consider that each eigenvector is of the form [ 'x', 'y'].
So you only use the 'x' parts for equating x(0) = 3.
For this example it would be...
Now do the same for y(0) = -4 and you should have 2 equations with 2 unknows ( and ) and you should be able to solve it from there!