Most internet readers do not preserve spaces at the beginning of lines so matrices are very hard to read.
It is much better to use LaTex:
Double click on that to see the code.
Also I used the capital Y because it would be very bad notation to use the same symbol for both the vector function and the scalar function it is supposed to be "equal" to.
If we set then that matrix equation becomes
Which then reduces to the two scalar equations and with the intial conditions x(0)= -2, y(0)= -4.
Differentiate the first equation twice to get and from the second equation that is equal to -3y- 4x: . From the first equation so we can replace it by that: or . Since x(0)= -2 and x= y', y(0)= -4, y'(0)= -2.
To find the solution to the intial problem means, of course, to find the vector function (since x= y').
Since you have found that (NOT " " because y is a function of t, not x!), and