Nevermind, I figured out how to solve the problem using my old differential equations textbook. In case someone is curious, here's what I did:
I rearranged the terms so that like terms were on the same side:
[tex]\frac{dz_{1}}{dt}=z_{1}^{2}[/tex]
[tex]\frac{dz_{1}}{z_{1}^{2}}=dt[/tex]
I then integrated each side:
[tex]\int \frac{1}{z_{1}^{2}}dz=\int dt[/tex]
[tex]\frac{1}{z_{1}}+C_{1}=t+C_{2}[/tex]
Since both sides had constants, I dropped one, and I then used the initial condition of [tex]z_{1}(t=0)=x_{1}[/tex] to solve for C
[tex]\frac{1}{z_{1}}+C=t[/tex]
[tex]\frac{1}{z_{1}}=tC[/tex]
[tex]z_{1}=\frac{1}{tC}[/tex]
[tex]z_{1}(0)=x_{1}=\frac{1}{0C}[/tex]
[tex]C=\frac{1}{x_{1}}[/tex]
I plugged in C above and got this equation for Lagrangian position:
[tex]z_{1}=\frac{1}{t+\frac{1}{x_{1}}}=\frac{x_{1}}{1+tx_{1}}[/tex]
