Say that $\displaystyle {\mathrm d \over \mathrm d t}G(t)=g(t)$ and solve the exact equation. I got $\displaystyle xy+G(x)-G(y)=1$ after using the initial condition. From that you need to find $\displaystyle x$ and $\displaystyle y$ so that the terms in $\displaystyle G$ disappear.