📂Numerical Analysis

Implicit Differential Equations

Definition ¹

In particular, in the context of ordinary differential equations, a differential equation that is expressed implicitly as follows is called an implicit ordinary differential equation. $$ f \left( x , \dot{x} \right) = 0 $$

Explanation

Of course, if there is a differential equation like $\dot{x} = f(x)$ it can be written as $g( x , \dot{x} ) = \dot{x} - f(x)$, so when one focuses on the term “implicit” one typically considers the case where one cannot gather $\dot{x}$ onto a single side.

Numerical-analysis difficulties

Implicit differential equations are distinguished from standard ordinary differential equations particularly because one cannot directly apply solvers such as RK4 in numerical computations. For example, an equation like $\dot{x} = \sin \dot{x}$ gives rise to the difficulty that the $\dot{x}$ appearing on the left-hand side requires the $\dot{x}$ on the right-hand side to be evaluated first in order to be computed.

Note that solving an implicit differential equation is not the same as using an implicit method. For example, consider the implicit Euler method: to solve $\dot{y} = f(y)$ one uses an update rule like $y_{n+1} = y_{n} + h f(y_{n+1})$, but here “implicit” only means that to obtain $y_{n+1}$ the term $f(y_{n+1})$ on the right-hand side must be evaluated first. In the case of an implicit ODE, however, one cannot uniquely determine $f$ from $\dot{y} = f(y)$, so existing solvers cannot be used — the situation is the same.

Consequently, implicit differential equations are treated as differential algebraic equation, i.e., they are solved by imposing $f \left( x , \dot{x} \right) = 0$ as a constraint.