Wow. I have plots. I need to learn some of the intricacies of Maple. Thank you everyone!

@ecterrab 

Ah! Thank you. That helped immensly. I did execute the procedure; however, I didn't include it in the post as it yeilded the functions of u and v that I wanted. The C's are functions of u(t) and v(t), and Gamma of just u and v.

I made the following changes:

inits:=u(0)=1, D(u)(0)=1,v(0) = 1, D(v)(0)=1;

sys1:= D[1,1](u)(t)+C[1,1,1]*(D(u)(t))^(2)+2*C[1,1,2]*(D(u)(t))*(D(v)(t))+C[1,2,2]*(D(v)(t))^2=0, D[1,1](v)(t)+C[2,1,1]*(D(u)(t))^(2)+2*C[2,1,2]*(D(u)(t)*D(v)(t))^(2)+C[2,2,2]*(D(v)(t))^2=0;

L:= dsolve({sys1,inits});

Now the output is just " L:=   " with no error message. I did not know what to search online to find this error, I tried searching for no outputs, but only found questions on suppressing the outputs.

I am aware that the geodesics package exists, but my advisor wanted me to write the code by hand so that I understand the material better. Truthfully, I may use that to compute the geodesics when all is said and done. Thank you for the suggestions.