DEqn1(t) =
In this case, the second input to your ODE function will have four elements. Using a slight variant (to avoid using y in two different ways) of the notation from that documentation page, vec′=f(t,vec), we have that vec is [x; y; u; v]. [The order doesn't really matter; you could have vec be [y; x; u; v] as well if that's more convenient. As long as you keep the same order throughout your entire code it will work.] That means you need your function f to return [x'; y'; u'; v'] (which is vec').
What is x'? From your transformation, that's just u which is the third element of the vec input to your function f.
What is y'? Again from the transformation, that's just v which is the fourth element of the vec input to your function f.
Similarly, you can compute u' and v' using the elements from vec as you've done mathematically.
Here's the general outline, you just need to fill in the FILL_THIS_IN sections. Then use this function when you call the ODE solver.
function vecprime = f(t, vec)
% Unpack vec into its components, so the expressions for the *prime variables
% will "look like" the mathematical form of the ODE system
x = vec(1);
y = vec(2);
u = vec(3);
v = vec(4);
% Define the components of vecprime
xprime = u;
yprime = v;
uprime = FILL_THIS_IN;
vprime = FILL_THIS_IN;
% Pack the *prime variables into vecprime
vecprime = [xprime; yprime; uprime; vprime];
end