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A partial differential equation (PDE) is an equation that includes differential(s) of one or more variables and implicitly stipulates that the equation holds when its other variables have a differential of zero. This is useful for the case in which variables are independent, i.e., when one does not necessarily vary with the other. This is in contrast to a total differential equation that does not stipulate this, i.e., the specified differential of a variable might depend on variation of the other variables.
Both are examples of differential equations, equations that include terms based on how much one variable is changing in relation to another. The corresponding derivatives are referred to as partial derivatives and total derivatives. A differential equation of just one independent variable is an ordinary differential equation or ODE. In written form, a partial derivative is indicated by use of a modified d (somewhat like a backward six: ∂) rather than a normal d. Example written partial differential equation:
∂x/∂t = 5 + ∂y/∂t
Example written total differential equation:
dx/dt = 5 + dy/dt
(∂x/∂t and dx/dt indicate the ratio of changes in variable x to changes in variable t. Similarly for y.)
Indications of position (over spatial dimensions) and time are basically independent (in a non-relativistic scenario), thus PDEs are useful and common in modeling functions on space and time (written in terms of coordinates). Such independence also can occur in equations built around variables related to such coordinates, such as velocity and momentum. As such, PDEs are ubiquitous in physics.
An example of the type of problem that can be stated as a total differential equation is a gas law (equation of state) written as variables indicating temperature, pressure, density, and volume, in cases there are dependencies among them. Non-astronomy scenarios include modeling economics, some principles that can be stated in equations with such cross-influences.
Solving a total differential equation often involves rewriting it as an equivalent partial differential equation, adding additional terms indicating the effects of the dependencies. Thus the total differential equation can be seen as giving a more compact statement of some aspect of a scenario, and may be considered a step toward forming the partial differential equation (the latter of which may imply an ability for one variable to change without another changing, that isn't actually possible). Obviously it is useful to understand the difference between the two types of differential equations, and to form equations using a consistent set of assumptions, with careful attention to notation.