This is a first order differential equations since it only involves the first derivative of y. It looks to me like it would be very difficult to solve!
This is a "system of linear differential equations". Unlike the first, this is fairly easy to solve for y
1 and y
2
This is a "third order linear non-homogeneous differential equation with constant coefficients" (the numbers in parentheses indicate derivatives). It's "characteristic equation" is [itex]\lambda^3- \lambda= 0[/itex] so has 0, 1, and -1 as characteristic roots. Once you have found the general solution to the associated homogeneous equation (
y(3)−y(1)=0), you can look for values of A, B, C and D so that
Ax+B+Ce2x+Dcos(3x) satisfies the entire equation and add that.
I'm very concerned that you need to "learn" these for an exam and don't know what kind of equations they are. That's peculiar.