In practice, the initial value problem is often solved numerically.

In this case, the equations do not possess a general solution, and usually must be solved numerically.

This is a transcendental equation in a and must be solved numerically.

This is easily rewritten as an integral in closed form, although one that can only be solved numerically.

The differential equations for the geodesic can be solved numerically.

This equation is implicit in Δf, and must be solved numerically.

And the equations of motion of the rise are solved numerically.

Shape optimization problems are usually solved numerically, by using iterative methods.

Often however, the equation must be solved numerically.

This is a dynamic system that is solved numerically.