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 $(x,y)=(0,0)$.
It is not hard to show that, up to a rotation, each geodesic can be parametrized in the form $$ (x,y) = \bigl(r\,\cosh t,\ \sqrt{r^2+2}\,\sinh t\,\bigr) $$ where the constant $r\ge0$ determines the closest approach of the geodesic to the origin. The element of arc length along this geodesic is then found to be $\mathrm{d}s$, where $$ s = t + (r^2{+}1)\,\cosh t\,\sinh t. $$
Now, the Jacobi fields split into the tangential Jacobi fields, which are spanned by $$ J_1 = \frac{\partial}{\partial s} =\frac{1}{(1+(r^2{+}1)\cosh 2t)}\,\frac{\partial}{\partial t} \quad\text{and}\quad J_2 = s\,\frac{\partial}{\partial s}, $$ 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 $x=0$. In fact, the more general family of complete, conformally flat Liouville metrics $$ g = \bigl(a_0 + a_1\,{x_1}^2 + \cdots + a_n\,{x_n}^2\bigr)(\mathrm{d}{x_1}^2 + \cdots \mathrm{d}{x_n}^2), $$ where $a_0>0$ and $a_i\ge 0)$ for $1\le i\le n$, has the property that the Jacobi equations on the geodesics of this metric can be explicitly integrated.
Remark: For more information about the metric $g$, in particular the explicit formula for its distance function, see this answer of mine