What distinguishes explicit methods from implicit methods in numerical integration?

The correct choice highlights the fundamental difference between explicit and implicit methods in numerical integration. Explicit methods are designed to compute the state of a system at the next time step directly from the current state. This computation is straightforward and often involves simple arithmetic operations, which makes these methods efficient and easier to implement.

In contrast, implicit methods require the solution of equations that include both the current state and the next state. This often means setting up a system of equations that must be solved simultaneously, which can be more complex and computationally intensive. This difference is critical, especially when considering stability and convergence of solutions. Explicit methods can be more limited in terms of stability, particularly for stiff equations, while implicit methods offer greater stability under certain conditions.

The other options do not accurately capture this key distinction. For instance, the assertion that explicit methods require more computational resources does not hold true generally, as explicit methods tend to be less resource-intensive than implicit methods. The claim that implicit methods are always more accurate is also misleading, since accuracy can depend on various factors including the problem being solved and the specific implementation. Lastly, while it's true that explicit methods may have limitations, they are not restricted solely to linear equations; they can be applied to both linear and nonlinear problems.