A particularly simple non-homogeneous example in which one can explicitly integrate the Jacobi equations is the complete metric on $\mathbb{R}^2$ given by $$ g = (x^2+y^2+2)\bigl(\mathrm{d}x^2+\mathrm{d}y^2\bigr). $$ It has Gauss curvature $K = -4/(x^2+y^2+2)^3<0$, and, visibly, a rotational symmetry about the origin $(x,y)=(0,0)$.
Now, in fact, it is not hard to show that, up to a rotation, every geodesic can be parametrized in the form $$ (x,y) = \bigl(r\,\cosh t,\ \sqrt{r^2+2}\,\sinh t\,\bigr) $$ where $r\ge0$ denotes the closest approach of the geodesic to the origin. One easily computes that the element of arc length along this geodesic is $\mathrm{d}s$ where $$ s = t + (r^2{+}1)\,\cosh t\,\sinh t. $$
Now the Jacobi fields split into the tangential Jacobi fields, which are $$ J_1 = \frac{\partial}{\partial s}\quad\text{and}\quad J_2 = s\,\frac{\partial}{\partial s}, $$ and the normal Jacobi fields $J_3= f_1\,N$ and $J_4 = f_2\,N$, where $N$ is the unit normal vector field to the curve and $f_1$ and $f_2$ are a basis for the solutions to the (linear) normal Jacobi equation $$ \frac{d}{ds}\left(\frac{df}{ds}\right) + K\,f = 0. $$ Using the above formuale, one finds that these can be taken to be $$ f_1(t) = r^2+1+\cosh 2t\quad\text{and}\quad f_2(t) = \sinh 2t\,. $$
Finally, note that these formulae generalize immediately to the case of the cohomogeneity-1 metric on $\mathbb{R}^n$ with the formula $$ g = \bigl(|x|^2+2\bigr)\,(\mathrm{d}x\cdot\mathrm{d}x), $$ since every geodesic in this space lies in a 2-plane through the origin $x=0$.